This is another subclass that has been not-so great since the beginning of 5e.

People don't play berserker barbs often because the benefit you get from frenzy is simply not good enough to counter the exhaustion you get afterwards. That bonus action attack is literally all you get from frenzy, none of the other berserker features use it or reference it at all. While that doesn't make it a bad feature, it's also not something I would consider taking levels of exhaustion to get at. Without exhaustion, berserker barb is one of the more offensive barbarian subclasses, it's also one of the most bland subclasses.

So, what can be done? Changing exhaustion is definitely an option, and I think that taking a look at the new proposed exhaustion rules from onednd might be beneficial, because they are far less severe than the current rules, but still impose penalties. The question will be whether or not larian is able to (or care enough about the viability of berserker barbs with 5e exhaustion mechanics) to do so.

Quote
EXHAUSTED [CONDITION]
While Exhausted (known in older books as Exhaustion), you experience the following effects:

Levels of Exhaustion.
This Condition is cumulative. Each time you receive it, you gain 1 level of exhaustion. You die if your exhaustion level exceeds 10.

d20 Rolls Affected.
When you make a d20 Test, you subtract your exhaustion level from the d20 roll.

Spell Save DCs Affected.
Subtract your exhaustion level from the Spell save DC of any Spell you cast.

Ending the Condition.
Finishing a Long Rest removes 1 of your levels of exhaustion. When your exhaustion level reaches 0, you are no longer Exhausted.