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 ...
The venerable Stephen Woodward recently published the design idea (DI) “Flip ON flop OFF” that converts a momentary push button to a classic push-on, push-off switch. Figure 1 is an attempt to go ...