OKay lets start with a 5e fighter... other then taking spells or magic items what feats or options give them social or exploration things?
Feats, magic items, and especially roleplaying and prestige through adventures and connections. Remember that the DM does not only have the rules at his beck and call to balance the game and give spotlight to all players more or less equally. This is why I don't consider it impossible to play story-orientated in 4e, as most of that still applies (although magic items less so, they are again extremely combat orientated in general), it's just taking in an awful burden of rules and constraints that will not even see usage.
battle master is what the base fighter should look like... give them those dice and option like the ones OUT of the PHB that add to skills, give them something like the jump extra far thing psywarriors get...
On this, I agree with you, the battlemaster maneuvers should apply to all fighters, with further specialisation down the line. The thing is that the Eldritch Knight is a fine class, and I think they still wanted a very easy option to play, hence the Champion. and most other fighter archetypes are fine too, if you take the above into consideration.
I still didn't use grids for most of 4e, 3e and late 2e used grids...
I did not play late 2e (2e was awful rules-wise (but brilliant settings) and the black book were just totally unbalanced). 3e grids were fun at start, but they made them so mandatory and rigid in 4e that I really got the impression of playing fantasy chess.
I agree that you don't need grids for social even in 4e, as for exploration, it's a bit doubtful as, from memory, most of the powers that apply are framed in squares. But again, if it's to play gridless and not use the (mostly combat) powers, why saddle yourself with all that ?
Note that there is
@pemerton's approach as well, which is very much story-orientated, and which is very 4e technical and very much about combat, but he apparently plays a lot by ear and by adjudication, again ignoring a non-negligible part of the 4e rules.
and yes it is 100% a flaw. Battlemaster is why closer to what I want, but is too little. There is 0 reason why you can't have non magic extortionary abilities
I'm not sure what you mean here, though. Honestly, it feels rather fine, and if you want to burn most of your superiority dices to use things like commander's strike (which I think is based on a 4e ability) and rally, it makes you a great leader.
I agree that having a few more abilities would be nice, but they certainly can be house ruled in. And they can (and should) stay very martial and explainable.
For me, the great fighters that I would like to emulate are the swordmasters of the Wheel of Time, and there is nothing extraordinary (in the sense supernatural) to what they do, they are just really, really cool for being so good at what they do.