It's a conceptual problem. The berserker barbarian is conceptually the strongest warrior in the gang. He's the one that should not only be beating up monsters, but bending bars, lifting gates, and kicking in the doors. However, when he uses his basic, starting, subclass feature, he suddenly gains a substantial penalty to his strength checks for probably the rest of the day, that means he is almost certainly no longer fulfilling that role in the party. Going so frenzied that he wears out afterwards is fun and thematic. But the way it is implemented doesn't support the basic concept well.
I assume this happened due to the designers not considering how punishing exhaustion actually is, including how difficult it is to recover from without expensive magic or days of rest. You get all you hp back overnight--exhaustion is worse by far.
So it is a complete fail concept wise. Even if it were perfectly balanced mathematically from every angle, winning medals in the Balance Awards, it would still fail to represent the concept well enough for me.
But, of course, it's not balanced all that well. I had a thread about this some time ago. Basically, you are better off taking a totem barbarian and Polearm Master. Now you can use your bonus action for an extra attack every round all day, rage or no rage, and take no exhaustion from it. (No, I wasn't comparing one character with a feat against another character without one, I just don't want to reproduce all the details.) You could also take wolf totem to increase damage through your allies with no penalties and no feat needed. (You need to be able to frenzy 3/day just to be able to reliably break even with the extra damage wolf totem can do with 1-2 non-optimized melee buddies.)
Its third failure is that it doesn't function as advertised. That feature looks like something you should be using all the time. But it only works as (some people believe it is) intended to if you save it for that one big fight conveniently marked for you at the end of the adventuring day. I personally don't think that was the intent, and if I'm right it's just more mechanical balance failure on the exhaustion penalties being too harsh.
After some exhaustive (heh) analysis, I settled on a house rule of allowing you to frenzy for free a number of times per day equal to your Con mod, and then start taking exhaustion afterwards. However, the suggestion in the OP of once per short rest limit and no exhaustion is actually seeming like possibly a better one. It maintains the concept of getting worn out (you can't do it again until you rest) but without the crazy hassle of 5e exhaustion. It also hits that 3/frenzies a day (assuming 2 short rests) you need to pull of to get good numbers.
Of course, the easiest fix is just to entirely remove exhaustion from the feature. It will make berserker clearly the best at doing focused weapon damage when raging...but that is kind of their thing. Mechanically, they are probably a little unbalanced at that point. The question is if they are enough unbalanced that many people will play one instead of their desired concept because the math is so attractive. So if someone wants to play a totem barbarian conceptually, but they decided to play a berserker and refluff it as a totem barbarian, because frenzy on every rage is just sooooo good, then it is too strong. If that almost never happens, it's alright.
EDIT: It just occurred to me that I undervalued the wolf totem's feature in my assessment, as it can kick in for any ally with lower initiative in the first round of combat. With that adjustment, I think straight up removing the exhaustion from the feature (just cross off the last sentence under "Frenzy") is within acceptable balance. The question to ask ourselves is, "If the PHB had been printed without the exhaustion penalty...would anyone be batting an eye at it nowadays?" I'm assuming it would be considered a strong option, but bear totem would also still be just as popular, as would paladin, battle master, etc.
