Part of that can be addressed on the player end. Once you know that unconsciousness means you're down for a long time, it gives more incentive to avoid falling unconscious, which makes it much less likely to happen.
Except it doesn't. You already don't want to fall unconscious, and when you do, it's not typically your fault. It's usually random chance with the occasional contribution of other character's self-interest or DM miscommunication thrown in.
To contrast, many players find that the default rules actually incentivize falling unconscious, since extra damage against you is ignored and the action economy means you lose nothing in the exchange.
Firstly, it's not really much of an incentive: 'losing nothing' presupposes that you have a healer who's initiative falls after the monster, but before yours, and between you the two of you don't have a better sequence of actions than those that lead to you taking damage and him healing you. You are always better off having some way to take the damage without anyone passing out.
Secondly, those players who have no fear of unconsciousness have DMs who are soft. It only takes one time for a foe to finish someone off (be it deliberate or accidental with an AoE) before you can see how terrible an idea it is to sit around on 0.
Taken to a more extreme case, the old rules where a new character would come in at level 1 would provide extreme incentive to avoid character death, which encouraged players to become more invested in their own survival.
Well, actually they didn't really. My memory of od&d and ad&d is one where I can't remember the name of any character I played, because becoming invested in a character was a futile exercise. Life was cheap, and characters survived more through luck than through having control of their destiny.
Which is kind of why the 'death on 0' rule was not a big deal - tons of things didn't even touch your hit points, and even if they did, having your hit points only get to 0 was really uncommon.
On a more practical note, if falling unconscious put your character out of action for eight hours, then nine-times-out-of-ten would just mean that the party retreats and suffers whatever sort of loss is associated with not continuing on for the day. It's not terribly likely that the whole group will just keep going like normal, while one player is sidelined.
9 times out of ten... if the DM supports the 5 minute work day. That's already a problem, and now you're saying it should just be assumed that the PCs have all the time in the world? And to what end? Now your 'penalty' is basically nothing, and falling unconscious is no big deal again.
Death at zero was the base rule in AD&D (2E, at least). Death's Door - where you can stay alive until negative ten, but you're super messed up until you can rest for a week - was an optional rule which was extremely popular. (Critical hits were another optional rule which was strangely popular.)
Deaths door was popular. I'm not sure that most people bothered with the "oh, by the way, you can't adventure all day even if you get healed" bit, or if they did, then it just got handwaved "yeah, sure, you rest up and can do stuff again".