+1
Currently, I often find myself indifferent to most consequences I suffer from fights, due to the knowledge that my characters will be completely restored as soon as the fight ends. As long as my characters have at least 1 health after each fight, it makes no difference how much damage they took, or how many abilities/spells they used, etc. But in a time sensitive encounter, I might need to go through more than one fight before resting, so I actually care not only about surviving the current fight, but that my characters stay relatively healthy by the end so that they aren't forced to rest after.
The introduction of more time sensitive encounters would make using a long rest a meaningful choice, without creating hard limitations on when players are allowed to rest.
D&D 5e is understood to be balanced around multiple encounters per adventuring day. Some mix of ~1-3 big battles and ~1-4 smaller battles with short rests interspersed, plus possible traps or environmental puzzles which further tax the party's resources. A party is not expected to nova all their most powerful abilities in any single battle; the challenge ratings of combats and the balance between classes are calculated with this assumption in mind.
If this guideline is
not followed, then yes we result in @Ferros's finding that the consequences of any fight are irrelevant.