I've got both. The .30-06 has been a go-to gun for years while the .270 made it to the range one time and has lived in the cabinet ever since. It does nothing the .30-06 can't do. As far as being a "flatter" shooter I guess the answer to that is yes but only by a few inches. Not enough to make any difference in most hunting situations but the advocates will swear up and down it's a game changer.
Shoot either one you like they will both drop game, really there isn't enough difference between the two to worry about. I don't "hate" the .270 but I wouldnt say I like it either. Why? Simply because I'm stocked up on loaded ammo as well as components for the -06.