It's a change in dynamics, but these two rules were not adopted at the same time: The first time we used a variant of the "skull marks/countdown to doom" rule was in 4e. I don't remember what the details were, but it did get reintroduced fairly early into our 5e game. The whack-a-mole aspect was much more pronounced for us in 4e, I cannot say definitively whether that was more because of the rule differences between the editions or an evolution of the group's play style. In either edition, though, it made going down something to be avoided a bit more, without penalising PCs too much or add extra tracking of any real consequence.
The KO rule was more of a gradual evolution/softening of going down, mainly to make it less annoying narratively, that eventually got codified maybe a year and a half ago or thereabouts. This is just something we strayed towards over time because it seemed to work for us, with less thought behind it. How much more it made the players avoid going down was a pleasant, if unintended, side-effect. The same effect can probably be achieved by making all enemies ruthless jerkwads who always make sure to finish their enemies off, but I just don't want them to be.
In any case, together these two rules made the players treat going down more as a real concern to be avoided and less as a standard procedure game mechanic.
I just let the healing potions flow freely (but expensively, giving them something to spend their money on) and allow short rests whenever it seems appropriate. Nothing outside of a long rest or some very expensive potions will untick those skull marks.