On the first commercial computer product I designed (for private aircraft), in the late 1980's, I had it test both RAM and ROM upon power-up, just because I saw that another avionics manufacturer did that and reported to the user that it passed. All it did is delay the user's ability to use it upon power-up. Predictably, the test was never failed, unless the entire product was non-op.
There's really no point in testing the RAM. If it doesn't work, you won't even be able to run the self test unless you can go with no variables and no stack use (JSR/RTS) etc.. It's not like a disc which may have bad sectors. The manufacturer tests every byte before selling the RAM. If there's any damage after that, it will be to the input or output circuitry that affects
everything, not just a random byte here and there. You could test to see
how much memory is installed, but there's no need to test every byte for that, just the ones on likely boundaries. You could also just tell it with jumper options, or just put the quantity in the ROM, and re-program the ROM in the unlikely event that you ever change the amount of memory.
EPROM is usually only guaranteed to hold its programmed data for ten years, so it might make more sense to test that. But if it's starting to go, the computer may not be able to run the test anyway. If it's on the raggedy edge, it may pass the test, and still fail to work after the temperature rises from warm-up. I just refresh EPROMs every 10 or so years, including on products I bought rather than made. I put a label on them telling when I refreshed them last. What makes refreshes easy is that you don't have to have the original files. Just read the contents into the programmer and program them back in.
Testing all of I/O would be quite complex, especially since the I/O ICs have so many modes of operation, timers, etc.. Testing one IC's I/O would require another I/O IC, and loopback circuitry. Then if there
is a failure, you have to troubleshoot anyway to see which IC it was.
I have
never gotten a bad IC brand new in hobby or in my prototyping work for my job. I have also been responsible for hundreds of thousands, possibly millions, of ICs in our products for our tiny company over the years, and to my knowledge, none of them were bad when brand new either. I usually go many months between customer repairs, and the few failures we do get have been infant mortality of relays (ie, if they make it through the first few months, they will probably last indefinitely), electrolytic capacitors of one particular brand, size, and value from one batch that started to go down after at least a dozen years in the field, on very rare occasion a monolithic ceramic capacitor or switching power supply controller IC that goes down for unknown reasons. We had one 65c22 almost 20 years ago that had an I/O pin get disconnected on the inside, undoubtedly a wire bond failure. Other failures, which generally show up and are caught before the unit is sold, are from human error in the assembly. Fortunately the SMT portion gets no errors anymore like the thru-hole stuffing did years ago.
There was a time that I accidentally let a screwdriver roll under my wire-wrapped
workbench computer on the workbench, and it shorted pins together and blew things, such that nothing would work so there was no way to run a diagnostic. I did not take the time to troubleshoot to find out which ICs were bad. I just replaced all of them and got back to work.