The 5 minute work day for lowbies is always going to be a thing because low level characters have incredibly limited resources. They run out of spell slots, hit dice, rages, and other abilities pretty quickly and have to take long rests to replenish and restock. Short of throwing in a Hackmaster style 20hp "Kicker" that's gonna continue to be a problem.I would argue that keeping characters up is more than a full time job in 5e, unless we assume that only one hit lands each round.
If we instead assume that a character might be hit 2+ times in a round, it becomes evident that a single healer cannot possibly output the amount of healing necessary to keep an ally standing when presuming less than ideal circumstances. The counter to that is that sometimes an ally might not be hit at all, but due to RNG we cannot presume to know in any given round whether it will be the former or the latter.
Your proposed solution does admittedly address this to a degree. However, I think you might need to examine how HD are gained if you implement it. The rapid scaling of HD (1 to 20) is not ideally suited to such a change IMO. I think that, without changing HD scaling, your proposal is likely to push low level parties towards the 5MWD (with the alternative being TPK resulting from Exhaustion penalties).
As far as "Exhaustion" I'm guessing you mean the persistent death saves. And yeah. That's a potential TPK issue for lowbies. Potential TPK issue for highbies, too. Anything that makes dying more likely is going to result in that issue.
The issue is: I see that as a net positive because it's meant to reinforce a different behavior from the Whack-a-Mole gameplay style. Can't have a stick or carrot situation without also having a stick on hand.