Having HP and injuries not be in conflict is simply a matter of redefining HP. Redifining HP also solves the problem with magical healing. This is a simple task. I can't see a player wondering about the thing you mentioned, because we can understand from real-life experience that it's different to be in "full health" with a chronic injury than without. We can also understand from real life that band-aid can't heal a severed limb - D&D healing spells have always been fairly specific of what they can cure (Cure Disease won't heal wounds, Cure Wounds won't heal Ability damage etc.), so this wouldn't be much of a deviation from D&D traditions. Injecting a bit of verisimilitude makes it trivial to explain, really.
Also remember the posed injury system is supposed to (partially) replace character death. The point being that the way D&D has historically handled death (at least before 4th Ed.) had all the caveats of injuries, except it was even worse towards low-level characters.
Equal-but-different still is different. Options 1) and 2) lead to vastly different chains of in-game events. The dilemma is preference - which PCs want one over the other? Character or player motivations can create notable amounts of drama over the choice. A problem here is your a priori assumption that there will be no conflict between these.
The underlined portion of your post is also wrong. If they are unaware
of a time limit, then they need to guess whether there is one. This is, in itself, one potential source of tension and conflict, because if there is a limit, then option 1) is clearly better, but if there isn't, then option 2) is. I've had great fun watch my players argue and bake their noodles over this dilemma.
If they are aware there's no time limit, then you are correct - they have no reason not to rest.
But overall, your guestion "What's the tension?" can't be answered completely on a system level. It's a matter of scenario design - a job for the GM or adventure desginer to set the table so the choice creates meaningful and interesting deviations in the story path.