I'd like to repeat Brian's conclusion:
TL;DR The myth of the Apple III's failure has been greatly exaggerated.
Yes, it's a bit like a fishing tale - it grows in size.
TL;DR#2 Nonetheless it was a major failure.
And unlike his conclusion of being a benefit to the Apple II world, it did set back all other Apple II development for years to come. The Apple IIe was crippled on purpose to give room for the Apple III.
A pretty good source to this is in the second part of an interview Wozniak in Byte issue 1/85 titled
The Apple Story Part 2, More History and the Apple III
Another source to consult is, as so often when it's about the APPLE (II), Steven Weyhrich's Apple II History site, here Section 7-THE APPLE IIE (And, of course, his great book).
In both places, he collected many of the issues that made the III the failure it was. All can be blamed to be rooted on management decisions, with performance and quality issues as direct results. They can be put into three categories:
Let's look at this in detail:
A Few Easy Steps Toward Failure
Like so often bad assumptions are a good base for failure. In this case it started how Apple management, led by Steve Jobs, did assume their market situation/product, where they wanted to go and what the proper future product would be:
Step 1: The Apple II is Dead, It Just Doesn't Know
In 1978 Apple started on two projects: A cost-reduced Apple II and the next big thing, codenamed LISA (*1). It was clear that the LISA project would need considerably more than a year - it ended being a good 4 years - so some stop gap measure was needed. At the same time, management felt that the hobbyist community wouldn't carter much to needed future growth, so a business focus would be needed, resulting in the idea of the Apple III.
The following year, 1979, was not only the year the Apple II really took off in sales, but its second half was marked by VisiCalc, the very essential killer application to the II. It became 'common sense' in marketing that the USD 2000 Apple II was sold as as accessory for the USD 100 program, as John Markoff put it in an Info World article. VisiCalc was, at the time, only available for the Apple II, no other machine, but it was assumed that this would change soon. Thus Management believed the 'hobbyist' Apple II+ might only have a remaining life span of less than a year.
The perceived reorientation towards business sales, due to VisiCalc (*2) made them believe that soon 90% were going to professionals (*3) while only the remaining 10% to hobbyists and schools. By the end of 1979, the whole company refocused 100% on the Apple III. Everything Apple II was outright cancelled.
Step 2: Designing a 'Real Good' Computer
Due the reorientation towards business computing, the machine had to get rid of everything hobbyist. As everyone knows, business computers are not only grey on the outside, they also do not feature any capabilities for games or alike. Serious, who would play games on an IBM-PC? (*1)
Then again, there were already lots of useful software packages for the II, so some way of running backward compatible software was to be provided. this meant that a 6502 CPU, as well as a Disk II compatible drive was a must. Understandable, but already limiting capabilities - especially due to drives with only 143 KiB capacity.
But instead of making the machine simply upward compatible, they opted to essentially build complete separate modes. A native Apple III mode providing all new features and a restricted Apple II emulation mode. Restricted as in restricted access, they did in fact add specific hardware to exclude Apple II software in compatibility mode from all improvements of the Apple III.
Even more, it did cripple the emulation to a level of an almost bare-bone 48 KiB Apple II+ with Disk II controller and a single serial (printer) interface. No Language Card, no 80 column display and no lower case characters. Switching to emulation mode was as well irreversible. The additional circuitry allowed no way to return, not even by reset. It really required a power-down.
At that point, quite a lot of Apple II software, especially all business-related, already required 64 KiB, lower case keyboard, and, of course, 80-column display. This made the emulation mode next to useless - even for games.
All software that intended to use more than 48 KiB, new video modes or the higher CPU speed, had to be rewritten to a different, incompatible OS. This included even high-level (BASIC) applications, as Apple III's Business BASIC was incompatible as well. All of this was intended, as Apple wanted to get rid of "all that hobbyist stuff". Applications should use the shiny new OS, real drivers and so on. An attitude pushed by Jobs that did not really resonate well with Apple users.
Not to mention that users had, of course, not only have to spend quite a lot of money on the new machine, but as well had to re-acquire all of their Software. Switching from an Apple II to the III was was like switching from Apple to any competitor. Not really a smart move, but a great help to make users decide, not much later, on the IBM PC. Except the ones that rather preferred to stay with their II+ and enhance them further - often way beyond what the Apple II ever could offer from the factory: 128 to 512 KiB RAM, 4 MHz CPU, high-res graphics and so on.
But sales to new customers were limited as well, as an Apple II with Saturn 128, 80-column card and keyboard modification was a way lower priced 'accessory' to run VisiCalc than an Apple III.
Step 3: Don't Feed the Zombie
Now, since the Apple III was set as the one and only product, there should be no more development on the Apple II, so each and every project, no matter how small, was cancelled. Including the cost reduced Apple II - after all, why invest in a new design that could not really generate any sales. Every business is going to buy an Apple III anyways.
Of course, reality became different. Other than assumed neither the hobbyist nor the educational market was fully tapped, even less saturated. There was huge growth and the Apple II a prime contender. A software base as good as their main competition, Commodore and Tandy, but way better expandable and thus better adaptable to new usage than both of them.
When it became clear, in 1981, that the III was not to be the instant success it was expected to be, while at the same time the II would stay (and grow) as sole profitable product, the cost-reduced Apple II was restarted. Great to go ahead, except, only under a strict mandate of in no way going to be better or even coming close to what the Apple III provided. So, base memory was still only 64 KiB (5), max expansion limited to 128 KiB (&) to max out at the lower end of the III line, and, of course, no larger floppies or, heaven forbid, the new 5 MiB Profile hard disk. Even 80 col was not included in the basic configuration, despite a separate board turned out to be more costly.
So while the 1983 IIe was a nice improvement over the II+, it was crippled by design with a sole intention to increase profitability for it's remaining lifetime to generate income until the III would take off.
Heck, even the double high-res mode which came hardware-wise for free was only included in a second motherboard revision.
Step 4: Designing the Crate Before Plucking the Fruits
It is common knowledge that the Apple III used cheap sockets to cut cost to make up for overrunning design cost - except, it wasn't exactly like that. The sockets weren't overly cheap nor did they cause the main fault.
Te real issue were general hardware problems due an overly complex mainboard design. Well, not complex because of the hardware included, but because space was artificially limited. The Apple III case dimensions were fixed early during the project - in part to get casting done quick - with no chance (or willingness of management) to change that. This restricted the size of the main PCB. To squeeze everything in, they had to resort to, for the time, extremly undersize traces. It's said that Apple hired three different external PCB design companies until one could finally create routing (*7) that made everything fit onto one board.
The issue was complicated by Jobs insisting that there should be no fan. He wanted a quiet machine. A decision that should haunt the machine soon (*6).
Step 5: Rush to Market is Always a Good Idea
The Apple III was already behind schedule, so Apple produced a first batch of 1000 boards and machines and delivered them ASAP as demonstration units to dealers. Of course, these boards had huge stability issues due their way too narrow traces not always being up to the current required, thus frying themselves. That is, if not shortcut or fried already during the wave soldering process.
And then there was the heat issue. Unlike the Apple II, there was no large interior space to dissipate heat and slots to let it go, but a quite confined environment. It is reported that, after some hours of operation, disk sleves of floppies started to melt within the drive. And that's where the story of IC sockets, falsely blamed to be too cheap, comes into play. Those were the very same sockets used in the II+ and other Apple products.
Machines switched on and off of course heat up and cool down. Like (almost) every known material chip pins extend and contract when heated and cooled, making them stretch and excerting a tiny force on their sockets, creating movement. This movement is defined by the material and the temperature difference vs. the force a socket holds them in place. Everyone working with old computers knows this - first action if a 'new' machine is opened is to press back all socketed chips that have moved over many years of operation. That usually repairs a lot of issues.
Well, for computers that cycle between 20°C and 35°C it can take many years until a single chip is 'spit out' of its socket, but when cycling between 20°C und 80°C, it can happen after a few weeks of operation, as the force is proportional to temperature difference.
To Apple management, it might have seemed less damaging to spread the story of some accountant trying to save a fraction of a penny per pin than admitting that the whole machine is a case of bad thermal design - heck, everyone understands that accountants have no idea about quality or technology at all.
Last but not least, there were huge gaps in QA - inbound and outbound. For example, the III was designed to have a clock chip from the very beginning, but these chips, delivered by National, had a high failure rate after just a few days of operation. Due to a lack of inbound QA and next to no burn-in before delivery, they failed at a quite high rate.
It's what Woz calls a 100% failure rate of all machines delivered (of the first batch).
Step 6: Closing the Coffin
Contrary to common assumptions, the October 1983 Apple IIIplus wasn't so much of a last attempt to put everything right, than a redesign required by the FCC to meet radiation regulations. Part of that requirement was that the machine had to have a new name. In turn additional shielding, new (shielded) connectors, an improved PSU and a new mainboard were added. User side improvements were a better, Apple IIe like keyboard and a new video mode for use with TV output.
Of course, none of this could really cure the damage anymore, as the basic issue of no real upward compatibility remained.
Then again, Apple's newest computer had just been released and everyone knew it had to be an incredible success, making forget that one little failure called Apple III and the forseeable phase-out of the Apple IIe. Everyone was expected to buy an Apple Lisa with its great, Pascal based OS and awesome Twiggy floppies.
*1 - At that time the design was quite different, based around a set of high-speed microprocessors, much more like what Jobs had seen at Xerox PARC, than the later, rather simple 68k machine.
*2 - Seriously, it cannot be overestimated what impression VisiCalc sales made on Apple management - way beyond the real role it played.
*3 - Real number of VisiCalc-driven sales might rather have been around 25%.
*4 - That is in case IBM would ever build one :)
*5 - Only because using 64 Ki chips was lower cost than building 48 KiB with 16 Ki RAMs in term of chips, sockets and motherboard real estate.
*6 - Since there was a new ASIC for decoding and a new dedicated memory slot, it would have been a non-issue to design the IIe to support multiple 64 KiB pages - maybe a MiB or more - something third-party expansion cards already did provide.
*7 - It was a time when automated PCB routing was still restricted to companies like IBM, while needing more space and more layers than hand routing.
*8 - One he repeated with the Lisa, the Mac 128 and even after returning to apple with iMac and Mini. But at least the first two did learn from the III's disaster, while for the later low(er) power design was already common place.