If you won't actually lose your character to a fight, how can it be considered such a serious fail state? How can it be a "death spiral" if there isn't, y'know, death at the end of it?Wait, who said you’ll lose characters more often than not?
By the proposed exhaustion rule, for example, you literally only need to be reduced to 0 HP six times in a single day in order to instantly die. Due to the amount of damage opponents can pump out (particularly at levels 1-3 when characters are fragile and the swingingy distribution is turned up to 11), you are nearly guaranteed to get multiple levels of exhaustion each day that combat occurs. The first level of exhaustion (essentially) guarantees you will always lose Initiative (because Initiative is a Dexterity check, so you have disadvantage on it). The second guarantees you won't be able to run away fast enough to escape any enemies hunting you down--and you'll need two whole days of rest to get back to normal. Three levels essentially guarantees you won't hit with attack rolls, and four halves your maximum HP, making death essentially guaranteed if you try to fight an opponent that can hit you with attacks or spells.
Realistically, one you hit exhaustion 5, it's game over because your speed is 0. Healing in 5e cannot keep up with incoming damage (as I mentioned before), so if you actually manage to get 5 levels of exhaustion, you're dead as soon as something can hit you. And because of that halved maximum HP, they need to drop you even less in order to kill you outright regardless of death saves. E.g. a level 1 Barbarian with 20 Con (rolled 18 for stat + taking Custom Lineage and the Tough feat) has 12+5+2 = 19 hit points. Half of that, even rounded up, is 10 (and IIRC the rule is actually that we should round down). That means an attack which deals 10 damage more than whatever your current HP is will kill the toughest possible level 1 character outright if they're already at 4 levels of exhaustion. Instant death, no saves, and there isn't even revivify available yet to fix it. For the vast majority of characters, that number is going to be closer to 5-6. That's not even going to require a crit; for many low-level creatures, all that they'd need is one solid hit while you're at less than full HP.
So...yeah. I said what I meant. Most combats are very likely to cost the party at least one member in the early game with these rules. And every party member you lose accelerates your own demise.
Under the premise of these rules, what's the point of the Fighter class? You're made for doing a thing that gets you killed. A lot. Like, getting into more than one or two fights in a single day is openly suicidal.