This is a pretty good point. I can also think of a scenario where a long safety battle causes both players’ clocks to be low so instead of trying to run out they try to “flag” the other player (make their time run out).
I’ll have to think about this more. That might add an interesting element to the game (like it is in rapid/bullet/blitz chess formats), or it might be too much of a change.
Something that comes to mind immediately is a delay per shot. It could be made very similar to today’s shot clock by using a 30 second delay with only 30 seconds on the clock to begin. This way the extension would be more like a time bank that you only start using once your 30 second delay per shot runs out. It would allow you to use the extension over multiple shots. I think something like this could be tuned to allow for plenty of thinking time in tough situations but increase the overall game speed. Something like a 90 second game clock (time bank) plus a 20 second delay.
This is different than how chess clocks work because the delay would have to happen every shot instead of every inning. In chess, ordinarily, the delay only occurs when the clock switches between players.