I am pretty sure I could. I know that I have run 4e with my homebrew cosmology, which has a non-evil plane of chaos and a non-good plane of law.Actually no, you can't use the great wheel with 4e, since you are lacking some alignments.
That ship sailed with 1e Dragonlance at the absolute latest (see "all the gods, even the 'good' ones, destroyed a city and wrecked two continents because one priest got above himself).And the mistake you are making is restricting alignment to this, because it's the very bizarre and unique 4e view, which is also (at least for me) one reason it failed, it reduced most things to "teams".
Nothing is "necessary". It is perfectly possible to play D&D for years with one small town and one dungeon nearby, with not a thought about cosmology.About half of the planes of existence in the Great Wheel are unnecessary.
My understanding was it was a bit of both. Someone came up with the pure alignment planes, someone else came up with a bunch of planes to represent afterlives and the homes of pantheons, and the GW as we know it was the result of mashing the two together (with a couple of extras to fill in the grid).Grid-filling is what a lot of early D&D was about. But specifically I think the original idea for the outer planes was to brainstorm as many after lives from religions and myth and legend first, and then after the fact they realized that they could be shoved into an alignment grid if they made up a few extra, rather than the other way around.
Anyway, the OP asked for this not to be an edition warring thread. So please, whichever you prefer, could everyone possibly say that without running down the other cosmologies?