And at 5 level the barbarian also gets 2 attacks and can frenzy for a 3rd.
The Zealot barbarian can just about equal the 3rd attack with Divine Fury. If they choose polearm master - it gets even better (the Berserker can't benefit from that as both use a bonus action).
The problem (for frenzy) is the very high resource cost - I can't (of the top of my head) think of any class ability that imposes worse.
The 6th level ability is extremely good and the 14th as well (10th level is a great ability in theory, but good luck having the charisma for it)- but frenzied rage is likely too high a resource cost.
This doesn't mean that the Berserker is useless, or even under-powered - as a whole. Just that Exhaustion is one of the nastiest conditions in the game and seems overly punitive relative to any other class.