It would be super simple to add the extension and the at-table button you describe.   The way I’d do it would be to leave the existing button intact, but add a simple phono jack (the headphone jack) .   Those jacks usually incorporate a simple switch, but you don’t even need that.   All you need to do is run a wire from the NO contacts on the ‘score’ buttons... Those are the big red buttons on the top of mine.   Just add your own button of Normally Open type NO and then run it to the table.   Pushing either the on table or on-box button would score.  Very simple.

Of source — adding two Bluetooth scoring buttons would be even sexier!  But, frankly, I LIKE the delay in game because I can catch my breath.... LMK if I’m clear enough...  Would be happy to expand

Question for you. In your schematic, how easy would it be to add an optional extension for each of the two player buttons that can be mounted underneath the table edge at each end of the table? As I see it now, this appears to be something that sits off to the side of the table, and someone (presumably the player who won the point) would need to walk over to the device to add their point. When we play, we quickly move from one point to the next. Walking to the device seems like it would add time between points, but having the external button available to each player on their end would be quickly accessible and not introduce too much of a delay in game play. Or did I miss something in the design that already deals with this?

Ha, superb! It's a chess clock for ping-pong :-D

That would be a good project — and much easier!   I’m working on a Soduku solution engine now.... One that does not rely on brute force or recursive decent.... One that mimics ‘human’ game play