I think the cap is there to prevent a voltage spike bounce back and false switch trigger or multiple switch triggers.
When I had a game with the caps on them I replaced them if needed.

I ordered some from Mouser or Digikey (cant remember ) but they are cheap and worth replacing.

Vibration and ceramic caps do not play nice together.....IMHO.

I read a boring tech article on caps and "thermal breakdown" was one reason for failure. Since the pinball caps see little heat from voltage its environment & installation is the only other source of heat. Its not an issue for a temp controlled iron but cutting short leads on a cap and using a hot iron could stress it. Similar to short diode legs and overheating it.

If you are replacing caps or other electronic components watch the heat and duration applied.