Yes. The amount of variance would depend on the physical composition of the capacitor. It's possible that their time constant has changed, and that they now take longer to charge.

If you have an oscilloscope, it may be useful to watch the capacitors charging from power-on

Yes, old caps can warm up and work, or warm up and fail.

Get the kit and replace all the old electrolytic and might as well do your IC sockets, as they can exhibit the same behaviors.

A bad capacitor might not be supplying the correct voltage or amperage, which could cause some strange behavior. Also, capacitors do sometimes bulge and leak, which can make a mess and cause damage.

Capacitors can be expected to last 10 years or so. Your game is now 35 years old, as are the capacitors. It's well past time to replace them.

There are capacitor testers available which can tell you with certainty if a capacitor is bad or not, but those cost around $150-$250. New capacitors, on the other hand, only cost a few bucks.

Here is a crash course on cap testing with those $150 meters:

https://pinside.com/pinball/forum/topic/vids-guide-quick-bally-driver-board-repair-bulletproofing

Sounds more like the connector, PIA, or sound generator has a flaky connection to me.

Changing caps is easier and probably needs to be done anyways, so i would start with that.

Is there a "best practice" to de-energize caps on pinball machines? I'd hate to short something and cause more damage.

Those old sound boards can have several problems:

1. Bad capacitors.

2. Bad solder joints.

3. Bad connectors.

4. Bad trimpots.

Yes, if the electrolytic cap is rated at 16 volts or less and it's a large value (like say 2,000 uf and up) It could hold a charge. Connect a 12 volt tail light bulb (or any 12 volt bulb) across that capacitor for a few seconds and that will safely discharge it.

Also check the 3 diode (1N4004) in series on the Vocalizer board.
Those puppy's run HOT.
Most of the time you will see the board is charred.

I had a diode gone bad because of a bad cap on the Sound Plus board.
Too much inrush current I guess...