I think what was meant with the 6-8 encounter day was, more than half of them should be skill based challenges and roleplay situations. Like you can have 2-2-2, to 2-3-3 in any combination either way, that should satisfy the 6-8 days.
It's not what was meant. 5-8 (not 6-8, not sure where the 6 came from) encounters/day refers to encounters which drain PC resources - i.e. hit points (and hit dice), spells, "per short rest" and "per long rest" abilities and so on.
Only a minority of RP or skill-based challenges are likely to actually meet these criteria, in my experience. Traps often will, if there are enough of them (not just one negated by a single roll). Trying to get across difficult and dangerous terrain may, if it's likely the PCs will need to cast spells to get past it and/or lose HP doing so. Social encounters though? They're extremely unlikely to in large part because it's so hard to cast spells in a social situation, and if a situation could devolve into combat, it's basically a combat encounter (just one you can avoid), and likely has either a zero resource cost (if avoided by Persuasion/Intimidate rolls together with good RP), or the same resource cost as a normal combat encounter. The binary nature of a lot of skill-challenge or RP stuff means that it's often hard to quantify in the 5-8 model, too, by which I mean so much of it results in a "zero resources used" vs. "lots of resources used" situation, whereas combat fairly reliably uses a moderate amount of resources.
I disagree with the OP that D&D 5E remains well-balanced at 1-3 encounters/day. At lower levels, it's not a huge problem, because the party has so few resources. It's thus very easy to think, particularly if you don't play above 5th or 7th level, that "This is fine". However, at, say 10th or 13th level? 1-3 encounters is a joke.
To challenge a party who are competent players, and understand how to manage their resources, with even say 3 encounters/day at 10th or so, you need to basically throw "Deadly" level encounters at them repeatedly (or close to it). This gets pretty old pretty fast, in my experience, and it still doesn't feel balanced, because it tends to be that the first encounter is a stomp, the second is a stomp but the players are concerned about getting low on stuff, and the third is precarious as heck (this ties in with what an earlier poster said re: a party with full resources being able to easily handle two deadly encounters).
I've got a lot of experience with this, note, because I have a party of 5-6 13th-level (actually 14th now) characters, and I've always found it difficult to build believable adventures with 5-8 encounters/day (outside of dungeon-type scenarios). Again, at lower levels, this seems to work fine - 1-3/day, and you don't even have to go that hard. Indeed 5-8 at low levels is often pretty scary stuff. But once you get up towards and into double digits, that has changed. Particularly if the PCs manage a short rest between the encounters. Whereas 5-8 feels really good at those levels.
I think any guidelines for 1-3 encounters/day are probably slightly pointless because the game just isn't built for it. A better approach, and one I'd consider with a new campaign where I was expecting to go with 1-3 encounters/day, might be to go towards the Gritty Realism optional rule in the DMG, though I think I'd go with the model where they can take a "break" for 1hr to spend HD, and maybe make the long rest be 3 days, not seven.
