My suggestion would be to look into the Powered by the Apocalypse ("PtbA") line of games.
There actually is a decent PtbA game that emulates 80's/90's action movies called "Action Movie World." But there are a lot of PtbA games, and most of them focus on one genre or set of tropes, so you can choose whichever one is emulating the kind of game you want to play.
Their main benefit in PbP is that, when you roll for things, the results are primarily narrative units rather than mechanical ones. So you can have a series of mostly freeform posts where everyone just narrates the cool things they want to do and then, when it's time to roll, you're not rolling to hit (and then damage) one person with one weapon one time, it's more like you're rolling to for the whole fight, or one dramatic turn of the fight.
Example:
In Action Movie World, there's a move called, simply "Violence." And when you commit violence you roll (using a different stat for up close or ranged), and the possible results are:
Best result (strong success): Choose 3 of the following.
- Inflict extra harm
- The target drops something
- The target is knocked down
- You make a mess
- Something explodes
- You hit a whole bunch of people
- You can escape or close in
Second Best Result(weak or mixed success): choose 1 of the following:
- You inflict harm but you take some, too.
- You inflict harm but are driven back.
- You inflict harm but a friend is hurt badly in the fight
And if you "fail" then the GM tells you what happens next.
So games tend not to focus on minutiae. There isn't a lot of 'roll to strike' and 'roll to damage' and 'you lose 3 HP' and 'roll to climb a wall/pick a lock/charm a guard.' Failing rolls just invites additional complication.
There's another move called "Stunts" and it's used to do cool things. Pretty much any cool thing at all that an action movie protagonist would want to do.
In my experience, with a little of pre-game work to help define expectations, boundaries, limits, etc, this can all be done with a minimum of back and forth. You assumes that each character can do whatever it makes sense for that character to do, and only roll when there's real risk or drama on the line.
They're not perfect. And in my experience, their success depends a lot on the group being on the same page about what kind of game it is and what kind of stories you're telling together.
But with the right group, I've found they work with PbP very well.