Path of the berserker is not defined by Frenzy - that's just one part of the kit, and it should be tactical choice by the player whether a situation warrants frenzying or not. If it's just something that you use every single opportunity, then it's not a feature that you're making a choice about, as a player- it's just a thing that happens, and that's uninteresting. You might have three encounters a day and choose to rage for all of them, at level 3... but only tactically elect to frenzy once, on the final major fight of the day. A barbarian that frenzies at every opportunity will quickly wear themselves out and become a liability to the party, the same as a spellcaster who expends all of their spell resources as soon as possible and then runs dry when it matters.
Yes, and they still will with my suggestion. You still have more uses of frenzy than short or long rests, so you can not easily remove all levels of exhaustion. You simply do not have the silly situation of only being able to use the feature once.
You can not frenzy at any opportunity, you can frenzy once or the levels build up.
Sure, charm and frighten immunity is nice, but can be replicated with muliple races, spells or magic items. But yes, I should not have neglected that feature.
Even with a proper exhaustion mechanic, Berserker is more solid than you give it credit for. That's not to say that it doesn't end up looking pale alongside some of the newer, and increasingly power-creep-prone paths - it's the oldest path we have for 5e, and the newer paths are progressively more 'interesting'... but it is solid nonetheless.
We can have theoretical discussions about their viability all you want, but in the end, in tabletop, nobody is playing them. Why is that?