Is it true that ctrl+alt+del only became a thing because IBM would not build Bill Gates a computer with a button specifically for the task manager? Making it so that Microsoft had to develop a workaround.
The answer to the question as written is no. However, I can see where it came from.
When Microsoft developed Windows NT, they decided they needed a "secure attention key" (SAK). This was a key, or key combination, that was guaranteed to bring up the genuine log in screen. The reason they wanted this is because, any sequence that could be intercepted by a user program could be used to spoof the login screen and thus capture login credentials. The SAK provides a means to get to communicate with the OS which no user program can stop or spoof.
In 2013 in an interview, Bill Gates said they asked IBM to add a key to the keyboard for this purpose but, by the time Windows NT was released in 1993, IBM had lost control of the PC hardware business and was in no position to force through such changes. Thus Microsoft fell back on using ctrl-alt-delete.
No, CTRL-ALT-DELETE was a thing before there were tasks to manage. The original IBM PC used this key sequence to reboot the computer, in case the computer crashed or the user just wanted to boot some other operating system or application. This was implemented by the BIOS, so any operating system, like MS-DOS, that used the BIOS services would get this behaviour unless they did something to change or disable it.
The use CTRL-ALT-DELETE as a "secure attention" key was first implemented in Windows NT. Microsoft considered allowing users to easily reboot the their computers with a simple key stroke combination like this undesirable. On the other hand, they needed some way for users to be able to securely login with their username and password without having to worry about being fooled by some program that displayed a fake login screen. So they repurposed the CTRL-ALT-DELETE as a special key combination that no application could intercept and that would always bring up a secure screen that users could trust.
(MS-DOS-based versions of Windows had a different and partial repurposing of CTRL-ALT-DELETE. With Windows 3.1 it would bring up a screen where you could kill the currently running task, while on Windows 95, and 98 and ME, it would bring up screen where you could select a task to kill. On both versions pressing CTRL-ALT-DELETE for a second time would cause the computer to reboot.)
Microsoft had no need to ask IBM to put a special button on keyboards for this. Just look at the bottom of the keyboard you're using now, there's a good chance you can see a Windows logo key there. That's wasn't something IBM agreed to add to their keyboards, it was something Microsoft put on their own keyboards and encouraged other keyboard manufacturers to do as well. If Microsoft wanted a special button for this they could've easily had one without IBM's help.
Is it true that ctrl+alt+del only became a thing because IBM would not build Bill Gates a computer with a button specifically for the task manager?
As if Gates would have had a request for a special key in 1981, when the PC was introduced,for some software that wouldn't be published until 4 years later (1985).
Sorry, that story smells way like made up afterwards.
It's rather the other way around. At the time Windows was released in 1985 the PC-AT was the actual design, and it did include a special key for exactly this: SysReq - a new and special key to invoke system services of a multiprogramming system.
So the question is rather why Microsoft did not use the special key. One may guess it was to allow sales for existing PC users.
Making it so that Microsoft had to develop a workaround.
It isn't a work-around, but a definite hook intended to reboot, guaranteed to work in next to all situations. Back at the time when the PC was designed, having a Reset button as regular key wasn't uncommon. In fact, the Apple II, without any doubt a major influence in the PC design, used Reset as a next to normal key in this role - to bring the system into a defined state to enter a command.
That claim does indeed smell like it was made up later, and with no regard to real world history or even realizing what the original IBM PC was capable of and the configurations in which it was sold.
The original IBM PC (IBM's model number 5150) wasn't powerful enough for anything resembling modern multitasking. It certainly had no need for a specific key for a "task manager". For that matter, neither do modern PCs, even though Microsoft have been quite successful in encouraging keyboard manufacturers to include a specific key adorned with the Microsoft Windows logo on just about every keyboard manufactured in the last upwards of 25 years.
It wasn't even entirely uncommon for it to be used without much of an operating system as we think of them today. Boot games weren't as common as later DOS-based games, but they were hardly unheard of either.
Consider that the minimal configuration of the IBM 5150 had 16 KiB RAM and only a cassette port for storage. It didn't even come with a floppy disk drive unless you wanted one (and paid the premium for it).
Now, most units which were actually sold were ordered with one or two floppy disk drive(s), or possibly had those installed as add-ons soon afterwards (not to mention having a bit more RAM than 16,384 bytes). But the architecture didn't require it, and the ROM BASIC included support for loading programs from cassette tape. This is similar to how other, contemporary systems were used, particularly in home settings.
The choice of the Control+Alt+Delete combination to reboot the computer was made by David Bradley and Mel Hallerman, two IBM engineers on the PC project. Regardless of the specific choice, one could argue that there is a benefit to having a multi-key combination to reboot the system, as it is harder to hit multiple keys simultaneously by mistake. (In fact, Wikipedia discusses how Bradley's initial choice of Ctrl+Alt+Esc was a poor choice because it was possible to hit that combination by accident, especially on the original Model F keyboard.)
Unless something is done by software to change this, on an IBM compatible, the BIOS keyboard handler code will generally be what intercepts Ctrl+Alt+Del, and trigger a soft reboot when detecting it. Modern operating systems, including Windows 3.0 and newer when running in 386 Enhanced mode, OS/2 and Linux will generally override this and use their own code to intercept that key combination and handle it appropriately. (On Linux, it can be configured to do everything from nothing at all to initiating a clean system reboot or shutdown.)
Windows versions which trace their lineage to Windows NT use it as the "Secure Attention Sequence"; the decision to use Ctrl+Alt+Del for this was made because, when the issue came up for Microsoft (which probably would have been in 1989 or so), it was the only key combination that wasn't already being used by a shipping application and therefore the only key combination that could be reserved by the operating system for its own purposes while retaining compatibility with existing software.
It's also worth remembering that at the time, Microsoft was little more than another software vendor for the IBM PC (as well as many other platforms). They lucked out in that they were one of the vendors who delivered early operating systems (other choices, besides no disk operating system at all, were CP/M and the UCSD p-System), and in that MS-DOS, under the name PC-DOS, quickly became a dominant choice on the IBM PC. However, Microsoft was in a far less privileged position back then compared to five, ten or fifteen years later.
My points have individually been covered in other answers but I'd like to present my own summary here.
ctrl+alt+del was added by IBM as a 'reboot' command to the BIOS. This did not involve the OS (though the OS could override the BIOS action).
For DOS-based Windows, a multitasking operating system, unmediated reboot (say to recover from a single stuck app) seems like not a good idea, thus the OS intercepted it, and where possible allowed the user to terminate specific tasks.
Windows NT needed a secure-attention-key for orange book(*) conformance. Since NT also aimed to run DOS-based Windows apps, it would not be a good idea to use any key combination needed by some app (the S-A-K must not be interceptable by apps). ctrl+alt+del fitted that requirement.
(*) The orange book is more properly known as Trusted Computer System Evaluation Criteria, a late-20th-century US DoD document specifying requirements for computing systems to be certified as reaching specified levels of security. At the 'B3' level there must be a 'Trusted path to the Trusted Computing Base for the user authentication function' -- for NT, typing ctrl+alt+del ensured subsequent input was not intercepted by a rogue app (unless the OS itself had been compromised).
Control-alt-delete was well known from early on as a means of rebooting a computer that was much faster (by about 30 seconds to a minute, depending upon RAM capacity) than cycling the power. The primary usage was escaping stuck applications, but rebooting to change what terminate-but-stay-resident programs were installed was also common. Under DOS, there wasn't much need to exit a stuck program without rebooting the machine, but under Windows that of course changed, and thus the key combination shifted focus to the former meaning, though for awhile it also provided the latter.