Don't reach for Exhaustion, it is the wrong mechanic. It's an existing and underused track that seems like it would be a good fit, but reflection on the nature of it shows it is the wrong tool for the job, and inflict a long term issue on someone who is likely not at fault. And, it actively encourages a selfishness of play that I find opposite of what I want at my table.
D&D is a team game. A character going down is rarely their sole fault, it's a team failure. A tank that was too good at their job of blocking all the foes and took too many hits so they go down. Or a tank that wasn't as good and let a foe get to a low AC & HP member of the party. A cleric that prioritized healing one party member over another and it was the wrong choice. Or just bad luck on getting critted or some failed saves. Having some sort of detriment to whack-a-mole healing (which is intentional in 5e) I can see. But having it last all day (or multiple) affecting one player when they may not have had a lot to do with them going down is punitive.
Third, front line is a valid and needed niche. Being willing to get hit and be in the fray, even if it's a heavily armored cleric who isn't using melee, is a definite thing in D&D. But, as the ones getting attacked the most they are the ones most likely go down to when luck runs poorly. So you are punishing a player for making a choice to play a character that protects others. The melee battlemaster and the archer battlemaster are just as resilient, but one will be targeted a heck of a lot more.
We have a penalty that only goes away once per long rest, but frontliners can get it more than once a combat. They fall, get Healing Word, and then are dropped again. Healing from zero is intentional design paradigm, as the game is supposed to be fun and making players sit out as a penalty is boring, especially in combat which is mechanically heavy so it takes a long time. Yet the way it interacts with the designer-intentional heal-from-zero can make it stack up quickly. So they can accumulate and accumulate throughout an adventure, only dropping by the smallest amount (-1) on a day they completely avoid dropping at all.
So Exhaustion gives inappropriate penalties, for a punitively long period, to someone who likely doesn't deserve it, and unjustly penalizes a class of characters that protect others and therefore penalizes play that I want to see (teamwork and selflessness).
Sicne heal-from-zero is intentional and not something we want to remove, there's no reason to have it in the first place. So it's a bunch of problems, to "solve" an intended benefit. It's there's no upside and plenty of downside.