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PC compatibles in the 1980s were often advertised as having zero, one, two, or sometimes more "wait states". Zero wait states was the best.

Basically, the wait-states I am asking about are due to the main system DRAM being too slow for the CPU, so extra bus cycles were added to make up for this latency. This reduced the overall processing speed.
I'm not asking about cases where a CPU is blocked from accessing main system RAM by some peripheral doing DMA, for example. Obviously, that's a feature for improving performance.

But I don't recall this being an issue on comparatively-priced machines with 16 or 32-bit Motorola processors, and running at similar clock speeds.

What was the cause of the wait states, precisely, and how come other low-cost home computers were able to avoid this performance problem?

It was an issue on all machines — wait states resolve any situation in which the part a processor needs a response from isn't yet ready to respond — but only in the commoditised world of the PC was it a variable and therefore worth putting in the advertising.

In the Atari ST, wait states are inserted if the 68000 tries to access RAM during a video slot (two of every four cycles), or when it accesses some peripherals (e.g. there is a fixed one-cycle delay for accessing the sound chip).

The Amiga differentiates chip RAM and fast RAM. Chip RAM is that shared with the coprocessors and in which the CPU may encounter wait states. Small Amigas like the unexpanded A600 have only chip RAM.

Conversely, on the PC processors scaled in processing speed much more widely, the underlying reasons for potential waits were much more variable, and one manufacturer would likely do a better job than another. So it warranted boasting about if your machine has a good number rather than a bad one.

The DRAM chips used for memory needed a certain memory access cycle length, for example 1000ns. Also CPUs needed several clock cycles to perform a memory cycle, so for example a 8086 could take 4 cycles to access memory. If the CPU is running at 5 MHz, the memory access takes only 800ns which is too fast for the memory. Therefore one wait state is needed to get 1000ns memory cycle. Lowering the CPU speed to 4 MHz would allow it to run with zero wait states. Basically wait states were needed because memory speeds were slower than what CPUs could access. Advertising does tell something about system performance. For example, if one system has 1000ns memories and another has 800ns memories, a 5 MHz 8086 is able to run at 0ws with faster memories and at 1ws with slower ones. In theory the 0ws machine can transfer data 25% more in same time than the 1ws machine can. Surely faster memories were more expensive so maybe it was important to advertise why two identical looking systems had a significant price difference.