When I DM-ed AD&D we didn't do combat with miniatures, usually either just a description or a really quick pencil drawing. I ran with facing, you could move to stop people from moving past you and all that fun stuff. It was more of a simulation than the tactical board game 4e is. A typical example would be if you played a Thief. You couldn't get behind somebody that where aware of you, they would just move back and/or turn.
I started with Holmes basic using miniatures and a grid in 1976. Began the foray into AD&D with the release of the Monster Manual in 1977 still using minis and a grid. AD&D DMG specifically mention miniatures and a grid as aids to play for the very purpose of eliminating ambiguity in combat. In fact, I can probably count on the fingers of less than a full hand the number of times I've played D&D (any version) without miniatures and grid. It was a number of years before I became aware that people beyond our gaming group frequently felt the game was
supposed to be played without them. Even if we hadn't started with them already available for our use we'd have made up something like them for that purpose.
Combat does get a lot more free-form and throwing a fireball when people where in melee range was not something you did with a light heart.
It was where I came from.

Oh, nobody TRIED to burn their friends with fireballs, but it happened a lot.
I am starting to wonder if it would be doable to run it a bit more like that in 5e? Instead of running it on a grid, just do it free-form and use push-backs and similar tactics to actually get behind mobs.
I don't see use of miniatures and grid as
needing to be constraints to otherwise "free-form" gameplay.
On the other hand, you describe what you are trying to do the same way you would describe it when role playing, instead of counting squares and doing small 5' pushes.
There is no practical difference to me between:
"He's blocking the passage preventing me from getting past? I shield bash him and run past. Do I need to roll or can I just do it?"
and,
"I can see that he's blocking the passage but I can also see there is room where I can push him aside. I use my shield bash ability, push him a square, and run to THIS square. I'll just take the AoO because doing all this will put me and the thief in flanking position."
except that one relies on DM judgement and fiat, and the other suggests
already fixed rules for what would likely be a fairly common situation.
Would you play in a game without miniatures where most of the combat is just described in words and sketched quickly on a piece of paper?
I have absolutely no reason to believe that it would be a superior approach to the game, yet have no issue with others doing so. But writing the rules that way leaves A LOT of people out in the cold who prefer NOT to play the game that way.
- Where simulation takes a front seat to gamist approaches?
When it comes to debate of, or even mere use of G/S/N terminology my eyes glaze over and make me want to just go watch TV instead of play D&D.