Because all those things happen to other characters too.
Powers manifesting differently in different people is a whole thing.
A class needs its identity by level 3, and if Supernatural Hero is part of the Fighter's identify, he needs to be supernatural by level 3.
Considering thats when Fighters get their subclass, it kinda makes that argument moot. The Fighter is supernatural if they pick the requisite subclass (nearly all of them save 1, may be 2).
And thats without considering the Supernatural nature of Second Wind and even Indomitable.