What is the rule of thumb for the sum of circuit breaker ampacities that can be installed in a breaker panel? (I know that we can put more than 10 twenty amp breakers in a 200 amp panel, but don't know what the upper limit is.)

The amount and size of breakers is irrelevant. You can fill a 40 circuit panel with 20 amp breakers, or fill it with 100 amp breakers, knowing you have 200 amp capacity. The issue is the total calculated connected load, that should not exceed 80 % of the Main, which is 160 amps.

I realize this may not make any sense, and this is not easy to explain without going through the entire Article 220 of the NEC.

Please come back with a more specific question related to your situation.