Some stuff I would do to fix 5e

[Ignimortis]

You don't need to codify it, give examples of stunts in the player handbook and examples of what is an appropriate skill and bonus for succeeding. I don't want a long list of stunts, I want to encourage players to invent it during play, and DMs to award the players for it.

Edit-
The player character is in a jungle, there is a spitting spider in the tree, they climb the tree to reach the spider. Then an ankheg bursts through the ground and tries to grab the wizard. The player character doesn't need a feature to cut a vine and make a swing-by attack, they need to think "how can I get back to the ground fast enough to attack/grab the monster all the way from up here? Wait.. this is a jungle, I'll Tarzan it!"

And then the GM sets a DC of 18 to do it, because they reckon that it's not something a regular person can easily do, but could still perhaps succeed at with some luck. And our intrepid hero, with their +7 or so Athletics, fails that check half the time. And that's perfectly reasonable by how 5e does such things.

Like I said, the numbers don't work out. It's not even about codifying those stunts into buttons, I don't want that either. But the sheer structure isn't there. There's no verisimilitude to it, because nothing in the game implies you're playing the kind of character who can do those things automatically or whose rolls are worth more than normal and therefore they should get like a DC10 and not fail and fall face-first if it still fails. Those stunts, if bolted onto 5e as it is right now, would be pulled out the arse, pardon my language.