He can spend the same amount on fighting... And I don't think it's ever been shown that he has to spend his extra feats on combat as well to stay competitive.
Again, most games to make it to the top, so we're mostly talking about 1 extra feat. And it's at 6th, where other classes are getting nice spells/abilities that help in combat, so, yeah, the fighter is likely burning his early extra feat to improve stats (STR, CON, DEX, WIS, all for combat purposes) or picking up a feat to improve combat output. If the fighter gets the opportunity to play into higher levels, sure, they can spread things out, but that's not exactly good design OR something most games are going to see.
Meanwhile other classes are picking up ribbon or primary abilities that enable out of combat pillar competence.
The fighter gets a few ribbons like the fact that he can complete many non-combat activities (especially in the exploration pillar) more quickly than others (using action surge) as well as the Champion getting Remarkable Athlete, BM getting Student of War and the EK getting spells. I also disagree, again, that it's an either or choice. The fighter doesn't fall behind (unless you can show me where someone has proven this out) if he devotes his regular feats to combat and only the bonus ones to non-combat.
The EK spell list is going to be almost all abjuration and invocation, with the opportunity for few non-blast/defense spells open to them. The BK student of war really only helps deal with combat pillar, although it does encourage a bit of social before kicking the door in (metaphorically). That doesn't occur very often - that you get a social encounter with a foe you plan to fight immediately after. The Champ does get a bit of boost on athletics, but that's not a skill that other's don't get better boosts on (looking at you rogue) or can't trivialize with spell selections. So, yeah, a handful of ribbion abilities that mostly don't even match up to other classes.
As for falling behind, yeah, he does. To the paladin that takes polearm master at 4th. To the barbarian that takes GWM at 4th. To the ranger that takes SS at 4th. If the fighter picks up Actor at 4th, he's behind the other fighty classes. That he now picks up one of those at 6th really doesn't help things, as the other classes have gained nice abilities to boost their combat effectiveness even more (spell slots, etc).
The fighter stays on par if he follows along with other classes at low levels. The raw combat power of the fighter is really locked up behind 11th+ play, when extra attack becomes decisive. Which is the crux of the OP, really, and a good argument. Again, this might not bother you, but it's there. If the fighter is going to be so focuses on the combat pillar that they only get options to branch if the DM is using an optional rule (feats) and then only a little bit and late in the game, they should be clearly the best at martial fighting. They really aren't, again, until higher level. For a class focused so narrowly on a single pillar, they need to be at least half again as good at that pillar as any other class. Fighters are on par in the combat pillar - slightly behind at low levels and then slightly ahead at 11+ and finally really good at 17+. That's not remotely great. Might not be a problem for eveyone, but I can clearly see it being a problem for some.