I appreciate the impulse to design some sort of new mechanical system that will find the sweet spot between metagame resource management and in-game narrative. That's a bit of a distraction, though, from what I consider the bigger flaw in your idea: players still largely control when rests happen and they will try to game them in a way that maximizes recovery and minimizes risk. This is the central tension underlying all of the hand-wringing over rest. Players want to minimize risk by maximizing rest while DMs struggle to find ways to swing the balance towards risk without seeming unfair or arbitrary.
The greatest strength of my house rule is its simplicity. There's no negotiation, no ambiguity, no jockeying to get around your new house rules. Players know exactly what rests will come and when. In a way, their risk is somewhat minimized. They don't have to worry about whether the DM's going to contrive to keep them from getting a short rest that they want. The schedule tells them when the next rest - next resource recharge - comes. So it's up to them to play smart or find clever ways of managing if they used resources too freely in previous encounters. As the DM, you're short-circuiting all of those tedious negotiations about whether the party can find a safe place to rest in the dungeon. And the players always have a safety valve if they want to recharge faster. But I've very rarely seen players use that safety valve to force an early long rest. They generally do their best with limited resources rather than take the increased difficulty.