Unless you have a spare SMPS on hand, I'd suggest replacing only the CMOS battery first. The battery inside a 500MHz computer will be on its last legs unless you've replaced it before. That's probably the reason for the checksum errors.
Also the power button could be sticking erratically in the 'on' position. I say erratically because, if it were stuck all the time, your computer will power down again after a few seconds. I've seen this happen before.
One way to check this would be to remove the power switch connector from the motherboard header and see if it still turns on by itself.
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