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.
A Berserker can comfortably wear two to three levels of exhaustion without really being ill-affected by it; they can take advantage to negate the attack disadvantage whenever they like, and they generally have advantage on the most common physical saves (Dex from danger sense, and Strength from Rage) anyway; being at 3 exhaustion for a berserker only really brings them back into 'flat state' with everyone else. It sucks, but it's not crippling, and wearing that penalty if you're really burning at both ends is a tactical choice the player can make for themselves as to whether it's worth it or not.
Don't undersell the other features in the kit: Berserker gains immunity to two of the most common ways that a barbarian loses their rage: charming (which either leads to them not attacking/taking damage, or else incapacitates them along with it), and frighten (which often leads to them not being able to make an attack, and sometimes also preludes what actions they are allowed to take); as much as it's a passive boon while raging, it's a potent one.
Intimidating presence is a little lacklustre, certainly, but it's not something that you generally use in combat - it's something that you tend to use before combat breaks out, or in social situations... and yes, it's Charisma. You aren't hitting them with your muscles, you're attempting to intimidate them with the force of presence that you and your muscles exude; it's a Charisma check through and through, and I'm honestly very tired of low charisma barbarians playing like they're high charisma, and asking for concessions to be granted to them for it. If you walk into a room and make a scene and flex your goods and roar, and you play your barb as comfortable doing that and as being able to really project their presence around a room... you're playing like you've got high charisma, and if you're doing that with your 8 charisma and expecting anything more than to be ignored or sniggered at, then that's as much of an rp faux pas as walking into a library and engaging with the local archmage about higher theoretics of planar travel with you're 8 intelligence. You can try both of them, but your chances of really doing it effectively are slim.
Retaliation isn't to be sniffed at either - it's a way of ensuring that the barbarian functionally always gets to use their reaction to attack something every turn, which is a really strong boon as well, considering that *most* turns go by without opportunity attacks being provoked, under normal circumstances.
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.
The concession that I would make would be to give them a feature that - as you suggest - allows them to remove 1 level of exhaustion on a short rest, but only Once per Long Rest, similar to the way wizards can only use Arcane Recovery once, no matter how many short rests the party takes. Remember that you already get one level of exhaustion recovered on Long Rest already - so that would effectively let the berserker Frenzy twice per day without any negatives at all, and only start to wear actual costs after that.
This all assumes that Larian are going to add an exhaustion mechanic for players - as others mentioned, that debuff on the bird has been there since day one.
