I've seen both types of barbarians in play and both are fine.
As has been mentioned above, frenzy generally only gets used once per day, usually a boss fight. In such situations, being able to deal out an extra attack really does make a difference. I've seen a berserker cut down swathes of enemies, then turn the boss into swiss cheese, all while the rest of the party was just trying to keep up. Basically, trading additional offense vs. the defensive benefits of bear totem.
I've only seen the berserker character frenzy more than once a day in two situations, and both needed it (the party really messed up, allowing their enemies to gang up on them). The first level of exhaustion generally only causes problems for initiative rolls and escaping grapples. The second level cutting speed in half slows the entire party. So exhaustion matters, but it's not a game breaker.
Again, as mentioned above, Mindless Rage can be really useful, but also rarely. When the berserker fought a sea hag, the hag's main ability was useless against the berserker. (The hag frightens enemies, then can cause frightened enemies to drop to zero hit points.) There was also a fight against a creature (I forget which one) that could dominate. Because domination is based on the charmed condition, the berserker was unaffected. A dominated barbarian turning on the party can easily create a TPK, but the berserker didn't break stride.
I think the main problem is that bear totem is just too good. It needs to be higher level than 3rd, or better yet, not a totem choice at all - something that all barbarians gain at a specific level. Still, a berserker can easily hold his own against other barbarian builds. The player just has to be a bit judicious about choosing when to frenzy.