I don't like the idea of having to sacrifice combat advantage to get the proper benefit of short rests. I think it should explicitly be an opportunity to rest and recover with no real immediate downside.
The reason I like having some level of recovery after each battle is because it means that each battle can be a bit more challenging. It's more interesting when the challenge in each battle is to "get through the battle alive" rather than "get through the battle without a scratch".
But it's fine by me for there to be eventual downsides like "running out of short rests", or getting fatigued because it's been too long since you've had a long rest. Like it'd be neat if at the beginning of the game, maybe you take a short rest after every encounter, but by the late game after the difficulty ramps up a bit, you're deliberately having to get through several encounters without a short rest in order to conserve them.
It's meant to be a tactical decision. You could also have the cleric use healing spells. But that costs resources. That's what D&D is essentially. Managing limited resources. Do you max out HP but use up spell slots? Do you cast a fireball now and hope you don't need it in the next encounter?
In old school, you rarely were at full strength. As a party, you had to decide whether to face the monsters head on or perhaps just sneak by. Or even negotiate. There was no true benefit to kill every encounter. But such nuance is hard to program into a game. It's best left to a DM. But I do find more satisfaction of defeating an encounter with less than stellar resources. It makes the victory that much more sweeter.