We have not, in general, experienced a whack-a-mole problem. That is not to say that we haven't had combats where the same PC has dropped to 0 hp in consecutive rounds, only to be revived by magical healing. We just don't have a problem with it when that happens.
First, nobody is actually dead when they hit 0 hp. Second, nobody is recovering in combat without some supernatural aid or the Healer feat, so it doesn't really run afoul of ludo-narrative dissonance to us. Third, it's not particularly different than D&D has been for the past 40 years.
Fourth, exhaustion as a mechanic is extremely punishing, and punishing the party for taking too much damage when they're ostensibly still in the middle of losing that same combat seems questionable. It's just asking the combat to snowball into a TPK. I'm sure some tables want that feeling where failure makes future failure more likely, but I don't think that's D&D's style. I also don't think it's particularly engaging from a narrative standpoint. Certainly not compared against the narrative of a ragtag team of adventurers rallying to overcome great odds and near tragedy. In general, we find being knocked prone, losing any spell concentration or ongoing effects like rage, often losing a turn, and consuming the healer's spell slot to be enough of a cost.
If I wanted to make healing costlier or make extended adventuring more dangerous, I would adjust how easily it is for the PCs to long rest, or adjust their maximum HP down after level 10 or so.