It should.
When you turn on the game and it's cold and start doesn't work , can you get into switch test mode right away ? Do any of the switches on column 1 or row 3 act up as well? Or is it specifically always just the start button ? If others are affected as well , depending which switches are acting up you can start locating on which side the problem is. If all the switches in column 1 act up , then the components for column 1 would have to be checked and tested, if all switches on row 3 are acting up, then components for row 3 are to be checked. If no other switches act up it would then mean the issues is around the start button itself or its wiring.
Solid state parts usually don't break in a "warm up and work" fashion. Capacitors , resistors could react that way if they were broken but it's also very rare. Corrosion can do this, but your board and the involved components do not seem to have any corrosion either.
Can you get into the test menus at all before it "warms" up ? If you can't either , I would start looking at the +12V which is common to all the switches. If this voltage takes time to rise , it would prevent start , and the direct switches from working right away.
I would also look at the 5v. I see you seem to have a board in between J210's plug. This tells me you've had issues in the past with power and instead of fixing the board correctly , a band-aid was added. If the PDB is clean and working properly , including all the connectors (most important) and a new cap job done (after nearly 30 years , it's becoming more and more important) then you should not need any "no reset" boards in your machine.
Something that people often fail to mention when they have issues like this is , how many mods have you added to the machine that is pulling down the 12v? You should not run a single mod connected to the power lines of your machine. Your mods should use a different power supply that is used just for the mods. The only acceptable mods would be a color DMD and GI/LED OCDs. Even if you think you have enough power left on the rails to power your mods, you cannot be certain that the mod will not add a lot of noise back on the power lines of your CPU board.
1) Corrosion
2) Voltage issues
3a) Chips and components
3b) Wiring and diodes of every switch in the same column / row.
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