On Barbarians: I don't mind magical barbarians- as the totem barbarian indicates, many barbarians come from an animistic culture. They gain their powers not just from being unusually upset, but also can call upon spirits to enter their bodies and allow them to go beyond their physical limits. If this means those spirits grant the barbarian special powers, well, some scholars interpret the myths of the Norse berserkers as them not just flying into a rage, but that people thought they could actually transform into bears!
So there's a lot of mysticism that can be attributed to the class, beyond them just being Mr. Furious. Barbarians can have supernatural abilities and that's perfectly in line with their archetype.
As for frenzied rage, well, forcing yourself to ignore the limits the brain places on the body is going to do some serious damage to a person. It's not that gaining levels of exhaustion is beyond the pale (to me, at least). It's that it imposes serious limitations on how often you can use the ability that don't line up with the idea of adventuring for days on end, or facing 6-8 encounters without a long rest.
This is why, when Rage was invented for 3e, the fatigue a Barbarian suffered was only temporary. The frenzied berserker needs some mechanic to remove those levels of exhaustion more quickly, and the subclass would be fine, IMO.