It does not say anything about official game balance requiring 6-8 encounters in a day.
The game is flexible in balance. You don't have to have any sort of precise "must do" requirements or the game falls apart. Still, when designing they had to make some assumptions because classes have different rations of at-will, short rest refresh, and long rest refresh powers. They have stated (and this is also in the books) that the baseline assumption is that 2 short rests in a day is typical. DM Basic Rules p.56-57 is the source for all of this, where encounter difficulty (easy, medium, etc) is explicitly
described and
defined based on the adjusted XP value of the monsters faced. Now, certainly you can have non-combat encounters that are challenges or drain resources! But this section that gives the metrics we are referring to doesn't happen to include them, so they should not be assumed as part of the 6-8 encounters. They may or may not be there, but the metrics remain even if they aren't.
The designers are going to judge class balance around their assumptions. Some players consider champions and warlocks weak because they
aren't using those assumptions, when both (sub)classes are fine if those assumptions are taken into account.
Unrelated to the above...
One comment I'd make is that exhaustion really is a pain. First off, you gain disadvantage on ability checks. This means your big burly barbarian is now no longer able to perform your party's out of combat acts of Strength, and it also means you are going to be easier to grapple and shove around in combat. That really makes you feel like a wimp. Secondly, taking more than one level of exhaustion in a day is just a generally unacceptable thing that shouldn't be assumed--just like any other effect that will carry over into the next day. You can't assume
greater exhaustion either--you may not have it, and it has an expensive consumed component. So as an assumption you really only get one Frenzy per day, and then become a wimp that other people can push around.
So, here are my current thoughts on how to improve Frenzy.
1) I think the subclass should grant a free melee weapon attack immediately after entering a rage--regardless of whether or not you frenzy. This addresses (what I consider to be) the serious problem that Totem Barbarians all get their extra rage feature every time they rage, while Berserkers only get one if they Frenzy.
2) It still feels weak to me. I'm thinking of a few possibilities.
a) You gain no exhaustion, but can only Frenzy 1/short rest. This would allow it to scale with the length of the adventuring day, exactly like the other barbarians.
b) Same as above, but if you Frenzy more than 1/short rest you gain exhaustion. This might be too powerful.
c) You gain one free Frenzy per long rest, and exhaustion for additional Frenzies. Seems a tad weak, but might be better than I'm thinking.
d) You eliminate the last level of exhaustion you gained with this feature when you take a short rest. This is almost identical to b), except that you are suffering the exhaustion penalty until you take the short rest. Balance-wise, I like this one the best. The problem with it is that it uses a new and messy mechanic. I hate those. I like my house rules to make the smallest tweaks possible and look like something that could have been in the book all along.