Early single-chip silicon CPUs like the Zilog Z80 or MOS 6502 did not have a multiply instruction at all. Was this because the technology did not exist at the time to implement it, was it too expensive or was there simply no need for such an instruction (like FPUs for Amiga power users, the majority of people could and did get by without one)?
Fast multiplier circuits as used today take enormous amounts of logic, far beyond what would have been cost-effective (or perhaps even possible) in the mid-70s for an inexpensive microprocessor. Even slow multiplier circuits (as would appear later on chips like the 6809, 68000 or 8086) use a fair bit of logic and would have very considerably added to the cost, perhaps forcing a multi-chip design with all the complications that entails.
The first lines of microprocessors were primarily targeted at embedded control applications where rapid multiplication is rarely needed, so that was likely a factor too.
You don't need it
Multiplying two arbitrary bytes together has limited practical value. (If you want to multiply by a constant you can hardcode the optimal sequence of instructions to do so.)
Obviously it would be nice to have but the expense isn't worth it.
In an arcade game... you basically never need to multiply a thing. To draw lines or circles, you can use Bresenham's algorithms. For nonlinear control problems, values from 0-255 are of pretty limited accuracy and you probably want floating-point anyway.
For financial calculations (or things like pocket calculators), you want to use BCD to avoid rounding errors. For spreadsheets or graphing programs, you need floating-point.
In microcontrollers, sometimes lookup tables are actually better for "almost multiplication" problems because you can put fudge factors in them to deal with responses of the physical system—motors or whatever.
Special mention goes to Elite, which managed to do real-time 3D graphics in 1984... now that could have really used multiply and divide instructions.
Slow multiplication implementations made with conventional ALUs and microprograms had another problem. There are a lot of machine cycles to execute a command. So much so that it becomes noticeable with intensive interrupt work. And for 8-bit microprocessors, with the exception of the case with the Atari 2600, working with interrupts generated by the graphics subsystem logic was very relevant.
A useful article about one of the first massively available single-chip 16 by 16 multipliers - here.
From the beginning of electronic computation, this was a common design decision when building a computer using minimal circuitry. The Manchester Baby, operational in 1948, had no multiplication hardware. Later, low end minicomputers such as the PDP-8 lacked hardware multiplication. For some, like the PDP11/20, there was an add-on peripheral for it.
It seems to be purely arbitrary (or pragmatic) choice of the designers, one of the main factors being the size of microcode ROM or PLA. As an example, I'll take soviet K1801VM1 CPU. Its latest modification, VM1G, does support multiplication. The only change is microcode, not even the size of microcode ROM or PLA. For the reference, look at this reverse-engineered verilog of the CPUs: https://github.com/1801BM1/cpu11/tree/master/vm1, specifically, cpu11/vm1/hdl/wbc/rtl/vm1_plm.v (two microcode versions in two modules).
Another example, though not an early one, is MC68HC05 embedded CPU. Being otherwise simplistic, it does support multiplication too.
Prior to microprogrammed/PLA-programmed processors, it took an enormous amount of control logic to manage a simple multiply (and forget about even trying floating point). Especially with early single-chip designs there simply was not enough chip space for the control logic.
With the invention of microprogrammed processors it became more practical to include multiply/divide and even floating point operations.
Only when graphics, etc, created a demand (and chip density improved) did hardware-ifying the operations become economically attractive.
(I worked on a microprogrammed processor for RCA in the early 70s. We were basically duplicating the 360 instruction set on an early LSI-based system, and it was a bear working out the microprogram logic.)
I want to boil down this answer to engineering decisions (as are many choices made in engineering).
A good engineer works with limits. One limit at the time was cost for producing the chip. More transistors leads to larger chip size leads to lower yield in the production and quickly increasing costs. In order to keep costs down you remove the "expensive" parts, such as multiplication and division circuits.
Did it work? Well, rumors has it that the 6502 was used in Apple computers instead of 6800 because it had a lower cost. So, yes, lower cost worked.
If we, playing with numbers here, assume that a 6502 with multiplication would cost 3 times as much -- would it be used? Probably not is my guess.