I dont think its the capacitors causing the issue, just giving you the symptom of the the fault appearing quicker\faster each time you try it.
There is no real way to test the PCB without a circuit diagram, with parts designed to be replaced rather than repaired these days, wiring diagrams for component parts are hard to come by.
However if you are happy to proceed with fault finding I would check all the inputs and outputs to the PCB, so can can find out when it first starts up what its it checking, and when it trips what part of the board is still live.
Another check would be disconnect sensors and start the machine up and see if you can get it to just keep the display on, and then disconnect the power, add another ribbon cable\spade terminal and start it back up and see at which point the fault is induced.
If that doesnt give you any obvious areas to investigate, I would next look at the PCB to see if you can see any burn or cracks on the board itself, blown fuses, a quick way of check would be by doing a continuity test with a multi meter.