I think it is a very delicate balance WotC has to maintain because the people who loathe "magic-like" abilities are very loud and very obstinate (just as much, if not more than the crowd favoring Balance).
Their issues usually center on the idea that it is ridiculous that a Fighter can only pull off a special move once an encounter. No matter the amount of fluff you generate to justify it, they don't like it. Dailies are even worse (and to be fair, they make no sense in-world for martial/mundane classes).
I think the middle ground that can be mined would be a small number of "special moves" that can be refreshed, like the ToB maneuvers. The real trouble is a) making those feel like feats of martial prowess and not like spells and b) figuring out a good refresh mechanic.
The cool thing is that WotC could easily release new Fighter Paths later on that focus on such things, so it isn't necessarily something that needs to be ready from the outset. To be honest, it might be better that way since WotC will have a better grasp of the system by then.
Baring ToB style maneuvers, what other kind of easily explainable and understandable "powers" could a Fighter have? He could definitely use an ability like Sure Strike (bonus to hit but does less damage). I could see him having a power to whack an enemy on the head to impose the equivalent of the stunned/dazed condition (or just disadvantage). Maybe a counter attack ability where if an enemy misses you by a lot, you get to make an Opportunity Attack (or a free attack) X-number of times per day? An ability where the Fighter just straight up gain advantage for an action. Channeling the Warlord, the Fighter could have "powers" to help his allies position themselves (free moves) and attack better (free advantage for an attack). There's a lot of things you could do with the Fighter when you start delving into team work, like distracting an enemy (sacrificing your action) to give an ally advantage on their attack or harassing an enemy so that they have disadvantage on attack rolls against everyone except you.
There are definitely things WotC could try here that don't set off that realism-ometer.