I'd guess you are looking at a cap failure in the original PSU. 386 era is what, early 90s. 30+ years is a good run. If a replacement PSU doesn't have a switch, you could always hook it to a surge ...

My 10 month old home-built system recently started exhibiting the following behavior. The machine refuses to power on using the normal power switch on the front of the case that's wired to the ...