My problem is, I 100% believe that healing should work in this general way...
...but I'm also aware that it is emphatically NOT how 5e was designed, all the way down to its bones, and if you tried to force 5e to work this way without adjusting numerous other factors, it would work rather poorly.
See, this whole thing is closely analogous to how 4e healing worked. In 4e, you have a number of Healing Surges, which represent (more or less) the total amount of gumption, vigor, tenacity, whatever you want to call it, that you can realistically draw on for the day. Every surge restores 25% (rounded down) of your total HP; characters normally can only spend 1 surge per encounter, using the Second Wind action, but Leaders, healing potions, and other tools or features could allow further expenditures. Fragile classes that didn't specialize in taking hits (e.g. "pure spellcaster" types like Wizards or "fragile speedster" types like Rogues) got few base surges, while classes that did specialize in being able to take the brunt of attacks (especially classes with strong physical defenses, like Fighter and Paladin) got a lot of base surges. You also got bonus surges equal to your Con modifier, and there were a few ways to get extra surges via feats and magic items that could boost the healing granted by your surges. (Dragonborn, for instance, had a feature that added their Con modifier to their surge value; for Con-specialized Dragonborn, this was quite a valuable benefit.)
But in order to work like the above, you have to design healing to work this way from the ground up. Healing surges put a very firm cap on daily HP restoration. If Healing Surges--or Hit Dice or whatever you want to call this resource--aren't actually baked in at the ground level and tested to understand the typical spread of results at the table, you're very likely to run into one of two degenerate situations instead. Either (a) surges/HD/whatever are unnecessary and simply become an exploitable bonus on top of other healing, or (b) they aren't enough to actually keep most parties going long enough to matter, turning the game into an exercise in frustration where you're constantly limited by an arbitrary-feeling cap.
4e did the testing necessary to make this system work. I don't really expect current-WotC stuff to be tested that well, and absolutely don't expect homebrew stuff to be tested anywhere near well enough to work. 3PP content like Level Up....might or might not, depends on a lot of factors.