Path of the Berserker is still viable. We have one in our party and he does obscene damage when he frenzies.
It's godly at lower levels (3 attacks at level 5....). But it does drop off a bit in later levels because the Barbarian can be shut down too easily, since he doesn't get any mitigating abilities (like ability to shake off fear). But that's not an issue with exhaustion. The exhaustion mechanic is OK.
1 level of exhaustion isn't really that bad, so it effectively becomes a 2/day mechanic (one during the day, one when you're sure you will rest).
The only change I'd make is perhaps that a long rest reduces all exhaustion if induced by rage, instead of just one level. After all you can't rage so bad that you tire yourself out for days - unlike starvation or whatever.