You only have total control over that when you are a DM - and 5 out of 11 PHB classes may technically be a minority, but it is a sizeable one.
It's hard to call it a design feature when nearly half the classes break it.
Levels 1-2 don't go by quickly at every table, either - you might take a month or two in real world time before reaching level 3. Moreover, it isn't just the fighter:
Swords bards can't use many swords.
Arcane Tricksters are merely tricksters.
Shadow Monks only have a shadow.
Ancestral Guardian Barbarians have no ancestors - at least no useful ones.
Rangers and paladins get to console themselves with spells at 2, at least.
AL characters must start at level 1, also.