With a couple more days of thought, I think my ideal mechanic looks something like this:
Limited attack/offensive/combat resources should be per short rest - but you should get few enough of these that a single hard to deadly encounter (or a few waves of easier encounters) can tax you. Offhand something like one spell slot per spell level - so a 9th-level wizard only has up to five spell slots at any given time. A dangerous fight will use the top two for attacking/buffing and the bottom two for defense - she might still have her 3rd level slot but she's effectively spellburnt.
A short rest (I like the refocus activity from PF2 as a model for this) should get back most or all of those slots - maybe just a number of spell levels a la Arcane Recovery, but simply all of them might work better. Other classes should be similar in terms of combat resources: Superiority Dice, Ki, whatever rogues get, all come back on a short rest. So unless it's waves of enemies, you're going into each battle with all your "ammo" ready.
Short rests shouldn't take a full hour, but should be long enough that they aren't guaranteed. 10-15 minutes to refocus sound right to me, especially if it's clear that you have to be doing that and not anything else. You're not refocusing while hiking or keeping watch.
Noncombat resources are on their own track: ritual spells (which should include most noncombat spells) would require special materials, lots of time, and multiple checks. Each ritual category (teleport, scry, resurrect, ward a location) should be it's own little subsystem, requiring particular components (which can include x gp worth of general ritual supplies, but most rituals should also have more specific stuff - ie to teleport you really want an associated object), specific checks (ie teleport wants a cartographer's tools check to land where you want to), etc. Rituals should generally be available to casters, but also anyone with the feat, and should essentially be a skill challenge every time, unless it's one you know you can pass because you've already worked it out (ie warding the camp). Rituals shouldn't outshine skills if they replace them (ie knock replaces thieves' tools, but is slower, louder, and more expensive.) Other nonmagical classes might get similar features, but aside from skill challenges and tier-specific skill check options I'm not sure what.
Special, signature high-level abilities each have their own rules: Divine Intervention already does, but things like a Warlock's 14th-level patron feature usually makes sense for having their own special rule, or a paladin's capstone transformation. These might be 1/long rest, but they should generally be flavor-specific and tailored to the class/feature. High-level spell slots might land here. A lot of high-level spells probably should.
Healing should almost always come from Hit Dice - ie healing magic generally works by letting you spend Hit Dice without needing an action of your own, and maybe adds a few points, but most if not all of the hp recovery comes from Hit Dice - and Hit Dice only return on a long rest. If you're out of HD, you're in a very bad spot and probably want to retreat form the danger zone right away, because you can't heal anymore. Lack of healing is the main attrition mechanic - one fight is a threat, but many fights is a new and different problem to be solved.
A long rest is a full heal-up, which means it only happens when you can completely recover. It takes days, requires safety, and completely restores you. The intent is for it to happen between adventures, and rarely during without special magic.