I started using a battlemat of some sort and minis or some other kind of character representation (coins, mini plastic army figs) within a year of picking up D&D (~1980). I drew 10' squares (2" scale) on a piece of posterboard, subdividing the squares into nine ~3' squares (back then, movement was in 3' increments). I used 2"x1" dominos to mark the walls and doors, and to serve as tables, thrones, stairs, etc.
Currently, we are using TactTiles and dry-erase markers, and real minis (most painted). I like using the battlemat and minis for many scenarios -- not just battles. I like using the battlemat for any RPG, not just D&D. I like the visual representation.
I have played a few times without a battlemat (all editions of the game, with and without AoOs), and I am certain that a battlemat is a boon to gaming. My latest experience without a battlemat was in a Game Day event a couple years ago. There was a lot of confusion between what the DM was describing and what the Players were picturing.
That some folks can play without the table-top representation of the battle scene, I'm happy for them. But I've never experienced a good (non-confusing) game without a battlemat at least for battles -- any game system. I don't doubt that *someone* could show me a good game without a battlemat at all, but I also beleive that game could be even better with a table-top representation of the action.
I like the table-top representation of the game action/scenario.
Quasqueton