Considering that Pokémon live in a world where the direct competition - other Pokémon themselves - can spam earthquakes, coerce brambles and vines to bind and strangle an opponent, breathe a fire whose burns cause "never ending pain", freeze the local municipality to below cold storage temp, corrode metal with their spit, read your dirty thoughts and make you feel the absolutest shame about it, and blast part of an island and a nearby airship at ~3 km distance with a superenergized beam of air...
Then it actually makes sense that, resilient as they already are, Pokémon can opt in to increase their healing ratio or healing scope from a lot of kinds of wounds, which is what most healing moves do and what restoring HP more or less counts as. Increasing healing ability temporarily costs energy to use (PP) and requires a good setup if you are left vulnerable, but speaking in terms of (Darwinian) evolution, the pay-off is exquisite.
After all, if you can recover to working capacity from a strong attack or a potent ailment and you also learned that your enemy can use it, then you literally live to fight better another day and acquire knowledge that for eg.: humans and animals in the real world is very difficult to gauge (in our IRL wild, if you are a wild animal and you suffer any moderately crippling injury, you are as good as dead).
So, what kind of things can Pokémon heal naturally? Presumably, whatever can be grown or developed naturally with resources; things like hide, fur, spines, blood and muscle tissue in general. So more or less what Sheep says. A broken tooth could probably be healed somewhat by empowered healing, but it's not going to end up like it was originally - while bone tissue in general can be grown, bone is a "once and done" thing in the general lifespan of an individual.
Where it gets interesting is with the inorganics. What *can* a Klefki naturally heal?