Putting healing and spells on different time tables doesn't accomplish that: the game will just go at the pacing dictated by spells, natural healing becomes irrelevant. Spells are the more powerful/important resources to recharge, in part because they can also be used for healing, if they're also the faster/easier resource to recharge, they become the only resource it's important to manage.
Tension can be heightened with (slightly - very slightly, you want to be careful about it) deadlier encounters.
As a DM-added, DM-arbitrated rule in 5e, I don't think balance would be a big consideration. A DM instituting such a thing would find his own balance that works for his campaign pacing, just as he finds a balance in encounter difficulty and spotlight time for the same. If it's worth complaining about D&Ds lack of realism or modeling when it comes to serious wounds and natural healing, and worth messing with the rather delicate balance among encounters/day, healing/day, and other daily and short-rest recharges, then it could certainly be well worthwhile.Yes. With the understanding that the wound has some mechanical impact, where the hit point loss (if it doesn't kill you) doesn't have any impact. It should be a trade off.
Balancing this would be really difficult, though.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.