Quartz
Hero
At low levels the fighter has his fighting style, which is a bonus to hit, ac, damage or protecting allies;
Not really. Notice, for instance, how the barbarian's +1 HP / level counters the duellist's +2 damage thanks to taking half damage when raging.
second wind and action surge.
Action Surge is a once per day ability that only gives one turn's effect. As for second wind, if they're out of armour, the fighter is going to be hit more. That and the half damage ability more than make up for it.
You missed the weapon master gaining the ability to crit more often as well.
That's not relevant at low levels and at the levels where it is relevant barbarians do boosted damage on a crit.
Barbarians from Lv. 1-10 aren't exactly raging every single combat, either, assuming the 6-8 encounters/day guideline from the DM basic rules. Even from Lv. 6 with 4 rages/day there's 2-4 encounters where the Barbarian isn't raging, meaning 33-50% of your encounters.
Yes, but they still hit first and have more HP. And when they do rage, and they can when they need to, they're going to take half the damage.
Not everyone plays a PC based on optimization
It's not a question of optimisation but of balance and fun. Why play a fighter when you're going to be so overshadowed by the barbarian?
As I said in another thread, ISTM that a champion-path fighter makes a perfectly serviceable drop-in character, but as a main character? I'm not so sure.