Umm, I'm not sure I understand the question. Regardless of the stucture, the only way to lose an encounter is to run out of HP. In an "adventuring day" paradigm, how do you lose an encounter?
In an adventuring day paradigm, the encounter isn’t the base point of balance. You don’t lose an encounter, you lose an adventuring day, by having to retreat and recover before achieving your goals. If you run out of spell slots or whatever and go back to town or camp for a rest, leaving the dungeon or other adventure unfinished, that’s a failure state. You did not complete the 6-8 encounter challenge, and as a result, something will change in the intervening time. The evil ritual is completed, or the dungeon gets reatocked, or whatever. In an encounter based challenge model, each individual encounter is designed to be a challenge on its own, rather than the challenge being getting through several relatively easy encounters without running out of resources. So, failing a challenge necessarily means running out of HP rather than simply calling it quits early.
And, frankly? Who cares? The odds of a group "losing" an encounter in any paradigm is vanishingly small. PC's almost never lose. That's the point.
In a 6-8 Encounter Adventuring day, you're going to "win" 6-8 encounters. In an Encounter based system, you're going to win encounter. It's not any different. And, in both paradigms, you stop when you run out of ways of recharging your HP. In 4e, it was when you ran out of Hit Dice, in 5e, it's when you run out of spells. 
Well, see, if you’re building a challenge model around the individual encounter and the party almost never loses an encounter, you’ve made an unchallenging game. Which I guess is fine for some players, but challenge is a pretty major gameplay motivation for a lot of players, so those players are likely to lose interest in your game. Meanwhile, in an adventuring day based model of challenge, it doesn’t matter if the players win most encounters because individual encounters aren’t meant to be challenging. Completing several of them in a row is challenging. And given the number of people who claim 6-8 encounters in one day is absurd, I’d say 5e is very challenging - most groups rarely if ever successfully complete a whole adventuring day. And it manages to be so challenging while also allowing players to feel powerful as they win most of their encounters. It’s a great model for high challenge with low lethality, which I think is a significant ingredient to 5e’s secret sauce.
In 5e, it's the casters that dictate pace more than anything. In 4e, the pace was dictated by the group and didn't really matter which classes you had on hand.
Back in my 3e days, when HP battery healing wands were so common, it was always the casters that dictated pacing. The casters are out of gas? Time to rest. We never ran into 5 minute working days because it took a lot longer than that for the casters to run out of gas.
Yeah, I agree that’s a big point in 4e’s favor over 5e. I don’t think that’s a flaw of the resource attrition model though, I think it’s a flaw of 5e’s uneven resource models. When martials are primarily at-will based and casters are primarily long-rest based, the casters control when long rests happen, which is not ideal. A better mix of resource models between classes, without going as far as early 4e did in giving everyone the same resources at the same levels, would be ideal. Late 4e, (from like PHB3 on, including Essentials, IMO) hit the sweet spot.
