I actually had similar thoughts when I was going through ideas for the shooting focused roguelike I was going to make, but this game mechanic totally changes how the game is played (although I believe it would reflect the actual FPS better, as you can constantly "strafe" as you shoot). Ultimately why I decided against something like this is because each turn would require a lot more input (both a move 'order' and a shoot 'order') and wouldn't have as straight forward an interface as DoomRL has now, which is one of the major things I admire, and one of the major reason why I enjoy the game so much.
(The compromise I decided on was that a significant proportion of the enemies in the game cannot fire every turn, but instead have a certain amount of turns they must take other actions, then fire again. For example, let's say enemy X can only fire once every 3 turns. Then a player facing this enemy in the open can keeping moving (and dodge) until the enemy fires, spend the next two turns shooting at it, then dodge again. This still has the shoot/dodge archetype while retaining the system of one action per turn. But facing a crowd of enemies in the open will be far less friendly, which I think is also a useful result.)