And this is with the Champion, well known to be the weakest Fighter subclass.
The Champion is supposed to be the best at pure combat and the simplest to play. I think you've missed that both classes can get attacks on their Reactions and Bonus Actions. These are not affected by Action Surge, but are affected by Fighting Styles, Reckless Attack, and Rage. This doubles the Barbarian's damage.
Let's take your 11th level numbers, both getting 8 extra attacks (Reaction & Bonus) over the 4 rounds:
Champ dmg per combat (w/4 rounds AS): 32*(.55*12.33+.1*19.33) = 278.86
Barb dmg per combat (w/Rage & RA): 16*(.78*15.5+.0975*28.5) = 237.9
Clearly favours the Champion. The Champion is supposed to be the best at combat.
Champ dmg per combat (no AS): 20*(.55*12.33+.1*19.33) = 174.29
Barb dmg per combat (only RA): 16*(.78*11.5+.0975*24.5) = 181.74
The Barbarian is better, and does even better as the combat goes on.
Let's rerun those numbers against a higher-AC foe. Say both need a 15+ to hit.
Champ dmg per combat (w/4 rounds AS): 32*(.30*12.33+.1*19.33) = 180.224
Barb dmg per combat (w/Rage & RA): 16*(.51*15.5+.0975*28.5) = 170.94
The Barbarian is not so far off the Champion.
Champ dmg per combat (no AS): 20*(.30*12.33+.1*19.33) = 112.64
Barb dmg per combat (only RA): 16*(.51*11.5+.0975*24.5) = 132.06
The Barbarian is pulling way ahead, doing one extra round's worth of the Champion's damage.
So let's take two average encounters where the Fighter does not use Action Surge and the Barbarian only Reckless Attack, and one boss fight where all are used:
Fighter: 2x 174.29 + 1x 180.224 = 528.804
Barbarian: 2x 181.74 + 1x 170.94 = 534.42
The Barbarian is very slightly ahead!
And we're pre-supposing that the Fighter uses Action Surge for dealing damage when it has many uses out of that: extra movement is the obvious one (remember that the Barbarian has Fast Movement), but there are many other uses. Remember that Action Surge gives you an extra Action, not only an extra Attack Action. I've been over this in other threads and this point has consistently failed to register.
Watch what happens when the Fighter does not use her Action Surge for combat. Maybe it was used to get out of a trap or save a comrade. For fairness the Barbarian has also no Rages left.
Fighter: 2x 174.29 + 1x 112.64 = 461.22
Barbarian: 2x 181.74 + 1x 170.94 = 534.42
The Fighter is way behind. I put it to you that using non-nova class features the Fighter should be a better fighter than the other fighting classes.
TLDR: A multiple-round Action Surge is well-balanced.