Thanks for all the responses.
I'm not here to tell anyone that they are wrong to enjoy the Berserker as-is. I'm not a super-power-optimizer either. But I like to understand my game balance, and I like to make sure my fixes are rare, simple, and do what I intend them to do.
First, I should say that I'm not convinced that immunity to charmed and frightened is as impressive as some opinions are saying. I've classified them into the "rest of the sub-class" that appears to be balanced with Berserker, as mentioned in my earlier post. Also, it seems to me that something is definitely broken if you take a sub-class and don't use its primary defining feature.
So I did some math. I compared the Frenzy feature to the Polearm Master feat. (I assumed the Polearm Master would get to use their reaction for an attack one in every five rounds. It is likely to be more often than that.) It is a good feat, but it can't be called broken without other shenanigans. I took it at 4th level, because I am not assuming a variant human. I assumed the barbarian begins with a Strength 16 and maxes it as soon as possible. I assumed the Berserker is wielding a greatsword and the Polearm Master wields a glaive. I compared the damage increase over a standard greatsword-wielding barbarian with neither Frenzy nor Polearm Master, presenting results as a percentage increase over the baseline. I figured this out for 8 different adventuring day configurations, and using Frenzy either 1, 2, or 3 times. This is what it looks like.
When Frenzy leaves the character more than a point or two behind Polearm Master in daily damage percentage increase I indicate it in red. When the opposite is true, I indicate it in blue. If they are very close, I leave it in black. This is an average over levels 3-20. (I checked to see if the values remained the same if 20th-level were removed from the equation. There were some minor changes in favor of Frenzy, but nothing sufficient to change the colors.)
I did not take into account the reduced combat effectiveness of the Berserker after the first Frenzy due to reduced initiative and speed, but this would drag their numbers down by a little bit more (turning most or all of those blacks red).
What I see here is that the only place the Berserker's Frenzy shines is if they have a very short adventuring day and Frenzy on just about every fight.
Only Frenzying once per day will leave them in the dust compared to a barbarian with the Polearm Master feat, and will have a pretty pathetic overall damage increase in general. It would make this feature utterly insignificant compared to other subclass options.
I submit that this is likely not what the designers intended, and that it is sufficiently imbalanced compared to the alternative as to be a legitimate objective problem.
In the final column, I've provided a comparison of how much damage is increased if the Berserker can Frenzy on every rage without suffering any Exhaustion. They have great nova potential in a short adventuring day, but classes are balanced around a longer day (just ask Mr. Warlock about his rests). It looks like they are in the range of 10% higher than Polearm Master for adventuring days of 6 encounters, and then start dropping down after that. So the question is, is the benefit in the final column too good balance-wise, or is it about right?