Honestly, just remove the exhaustion, make it so that you can frenzy a number of times per day equal to half your daily rages (rounded down) using this ability consumes a daily rage. At 11th level, all your rages are frenzied rages.
As it stands now, the ease of which enemies can stop your rage makes it particularly brutal when an enemy uses a crowd control spell on you when you enter your frenzy, instant level of exhaustion and you wasted a daily rage. For an archetype defining feature, that's bad. No other archetype has this kind of drawback to their defining feature. The number of rages you have per day is enough of a limiting factor for frenzy to keep it in line. 1 Frenzy per day until level 6, 2 per day until level 11, then a number equal to your daily rages seems fair, you took the damage archetype after all. Plus this keeps their number of attacks per round in line with a fighter at the level 11 mark, so it certainly isn't overpowered when fighters get action surge per short rest, and before level 11 its a small number of times per day compared to the fighter's reliability. Besides, fighter's also get maneuver dice for even more damage compared to the barbarian. The Paladin will always burst harder than you, the Fighter will always do more average damage, and both of them get fighting styles, but for those 1-2 fights a day in the early levels, you feel good about yourself. I honestly think it's insane that wizards decided to make barbarians the least damaging of the "fighting man" classes, I'm ok with totem being a defensive choice, I'm not ok with berserker being a trap choice.
Barbarians have always been my go to when I wanted to make a martial front-liner, unfortunately as it stands fighters just do it better. More damage with extra attacks and action surge, more options with maneuvers (which also add a hell of a lot of damage.) Sure, their mechanics aren't as flavorful, but you shouldn't need mechanics to role play your character raging. I could just as easily say my fighter flies into a rage as he runs at his enemy. I don't get a mechanical benefit, but story wise it's the exact same outcome. Thankfully most DM's I have had the pleasure to play with allow some house ruling to buff the berserker to a point that it's not a trap, so I've been able to play a fun, competitive, berserker. House rules truly make the berserker a fun archetype to play instead of a gimp.