The DMG itself says that you're expected 2 short rests to every long rest. This is not a "judging by the numbers" thing, this is something the DMG outright says is design intent.
I mean, rage can end prematurely. That's one reason you'd want more than three uses, even if your DM doesn't have more than 3 encounters per day.
But, we shouldn't get lost in the individual examples. There are more than just the number of rages per day. Monks gaining Ki, for example. There isn't much point in having 15 ki points if you only have 1 encounter/short rest. It's extremely unlikely a monk is going to burn more than 6 ki points in any given encounter. Well, I'm thinking Way of the Hand monks here. I suppose the elemental monks might be burning more with spells. But, at that point, if you're dropping that many spells in an encounter, why aren't you just playing a wizard?
Then there's things like the extended duration effects for things like Hunter's Mark. Again, not much point in having a spell that lasts for 8 hours if you're only going to have 2-4 encounters per day on average. And there are other spells like that as well.
6-8 per day also seems to line up pretty closely with how the modules seem to play out as well. At least, in our experience. Granted, I haven't played all the AP's, just Ravenloft and now the Giants one and half of Hoard of the Dragon Queen, but, from what I've seen, the whole 2-4/day thing just wouldn't do it justice.
Heck, I'm in the middle of MT Black's remake of Expedition to Barrier Peaks, and the group is going through half a dozen encounters before long rests pretty nicely. Works out quite well.
The problem with going with shorter days, and then expecting the system to back you up on this is that you would need to rewrite the entire Monster Manual and a good chunk of the PHB to make it work. The only way you can make the encounters dangerous enough is for the encounters to last longer. Instead of 3 rounds, each encounter has to last 5 or 6. But, then we're back into the grind complaints that plagued 4e. If the encounters go back down to 3 rounds, then we get the Rocket Tag effect of 3e.
In my mind, the 6-8 paradigm is a compromise between the two extremes of 3e and 4e. 3e you have pretty short encounters (round wise) but not very many while 4e you had very long encounters but, you could keep going for many more encounters per day. 5e is largely splitting the difference.