Hm. Thinking about this, to have this work, each individual action would have to be relatively minor. So giant HP pools or really weak attacks/heals, things like that. That's usually the opposite of what most people tend to enjoy (and would almost certainly pad out the length).
It'd be a game where you only achieve victory through innumerable small actions, with few to no discrete moments of "that was a turning point" or "that's when victory was achieved". A downside of this kind of game is also when you often then -- late game or late battle -- suffer through "I already know the outcome, but we're still going through all the motions until resolution completes". By
that metric, a lot of 4x games fit this: while early game each individual action matters. But late game, it's just clean up and waiting.
I do want to point out the opposite (ish): The recent game
Old World. It's a great spin on a 4x game where one of your empire's productions (beside cash, wood, etc...) is "commands" and they clear every turn and certain parts of your economy produce them [rather than other bits]. You're pretty much always strapped for these. You're never going to have enough to do everything every turn. If you're at war and spending your commands on military units, you won't have enough to keep up ordering your workers to do things. Or if you have a major dying-off in your governors/advisors/etc, you're going to be spending your commands filling the vacant seats rather than giving other more direct orders. And of course you can (at a pretty inefficient conversion rate) buy more for when you really need to, or even over-spend commands to push a single unit well beyond its usual per turn limit. And certain actions pre-spend future turns' commands.
It's
really elegant and a great departure from the norm. And, depending on your goals, it
may achieve some of the emotional goal you're looking for here. (Of course, doing the right things to produce sufficient commands is extremely important -- so it's not at all "action economy is not important" but is very much a different-than-usual action economy.)
EDIT: It's at a more zoomed-in level than most 4x games, so units can move pretty far on a given turn, rather than each individual square of move being critically important. But, during combat, since it's a 1-unit-per-tile game, maneuvering can get expensive and important. I still haven't won a game of it yet; and I've only played through 3 since it came out a month or two ago.
dtgreene: There are a few other games I've encountered, mainly JRPGs, where you can spend all of your MP (or similar resource) on a single powerful attack, but usually there's the alternative of using weaker but more MP-efficient (but less action-efficient) attacks.
Psionics in D&D 3e.