At some level what you mix as a resource and what you allow to pool as a resource is arbitrary.So, in the case of the paladin, I can completely see there being some level of divine connection (or whatnot) that is being depleted. Similarly for casters just having a certain amount of brain space to store spells.
The place that it gets me is when it is pushed off to the "martial" type things where it's described as exertion or effort. Story-wise, I'm having trouble with why maneuvers (say) wear one down in a totally different way than the soaking or dodging in the
non-meat part of HP does.
A fighter can once per short rest action surge and once per short rest second wind, but they can't mix and match them as a joint pool of things to do per short rest.
Conceptually they could and then it would be more of a tradeoff of opportunity costs whether you choose to self heal or whether you choose to double your attacks/actions.
You can come up with narrative reasons for things like the differences, taking a breather to restore your fighting spirit (second wind) is different than pushing your body to do more in a short burst. Forgoing a breather wouldn't normally let you push yourself to max capacity more in doing something like pull ups.