Doom and Half Life (and other such FPSs of its time like System Shock 2) seem to me like two entirely different genres. I think Carmack was wrong but you can still have fun with a simple shooter + puzzle game like Doom.
Given the lack of ammunition, and the overall design, I find it hard to believe that there's a way to play it on which a significant amount of time would be spent shooting.
Games don't have a minimum "bullets per minute" quota they need to fill to be considered a shooter. Sometimes I go on a melee spree in Halo, does that make it "not a shooter"?
SS2 lets you use projectile weapons as your preferred way of disposing of enemies. In my mind, if you choose that path, it could be described as a shooter. That isn't all it is, but it is one word that describes a particular way you can choose to play the game.
Side note: SS2 is probably my favourite game so I certainly don't mean to "degrade" it by implying it is "just" a shooter.
