I also have to say that I didn't understand your "accurately judge the terminal space of their deck" example, because I'm just not a good enough player to see the relevance.
And I still maintain that a line in the game log at the beginning of every Buy phase that displays remaining actions, buys, and coins would be really helpful both during and after games.
Besides Villa and that one Tournament prize, what is the action count at that moment good for?
Both of these things are answered by the same concept. "Terminal space" is one of many things I just made up, but then I saw a bunch of other people using the term and assumed it was clear exactly what it meant.
I defined "terminal space" as the number of terminal actions you can expect to play on your turn. If you're drawing your deck, you can usually calculate this pretty easily. It's useful because it's one of many metrics you can use to gauge the capabilitity of your opponent's deck.
Why is it a useful thing to know? The trivial example is that I play a Swindler and now I have the opportunity to give my opponent a village, a non-terminal, a terminal, a treasure, etc. and I want to know which one is least useful to them. If they are "over-terminaled" or "don't have enough terminal space" or "have too many terminals" then I want to give them a terminal. If they're being the "village idiot" then I'll just give them another village.
But knowing the exact capabilities of your opponent's deck is crucial to being able to properly play an endgame. If they have extra terminal space then I know they can add terminal payload to their deck and have higher capabilities that I now have to play around. If they don't have extra terminal space and villages are out, I know their deck is limited in its potential. It can even have insights into how well their deck is running, since some cards can be played for other benefits besides actions. It's hard for me to tell how many of their Squires are being played for Actions when that information could be very relevant (they played a Squire for buys they didn't use, but they could have played it for actions instead, so they have more terminal space than is showing -- this is why having lines in the game log when these types of choices are made is very important).
All of this is strategically useful, but probably the most common use case is that I just want a concise summary of my opponent's turn, and right now all of the information I need to get that summary isn't there if I didn't catch it on my opponent's turn.
If they played their whole hand, it may give you information, but if they kept a few cards in hand...?
You make a good point. Number of cards left in hand would be great to know as well.