The original game does in fact have great gunplay. For the longest time I kept ignoring the game on the assumption that it had come out too early in terms of technology and design ideas to be other than simplistic and repetitive. I gave it a proper try only a few years ago and I was taken aback by how good the whole game felt, from start to finish. What makes it so good is the flow or rhythm of the combat, but that probably depends on a really fine-tuned combination of enemy attack types, mixing different kinds of hit scan with various projectiles, their movement patterns, the way enemies complement one another and how that lends itself to switching between multiple weapons, the damage your weapons do, the spread, rate of fire, etc. Being limited to horizontal aiming is in no way a problem when the game is fully designed around that set-up, perhaps it even made Id put so much care into tuning the combat within the framework that was available.
Naturally, the level design, music (especially rendered on Roland SC-55), and the way atmosphere was conveyed with the little means the texturing permitted are also important. This is why Doom II wasn't quite as good, despite some interesting encounter design: the levels looked blander and more often halted the player's pace; the music also was coposed more for a sort of slower, methodical play. It's important to set up DOSBox right if playing that way and not to enable any additional features in source ports (like the standard lighting in GZDoom which makes everything look duller), they only spoil it.
Verticality isn't used to that great effect in most fully 3-D games anyway, but that's rather because they don't below to the same type of games as those nineties shooters. Amusingly enough, Id used verticality far better in Quake than it did in any of its later games. But the main problem with the development of shooters was that the whole approach to design simply changed towards the end of the nineties and all the possiblities of better engines were never really utilized to attempt to expand on that original approach. At least the new Id is trying with this new take on the series and it's something substantially different from the earlier multi-platform shooters, but it's put forward as a successor to the original game, which still literally holds up, it's still one of the best shooters, not for historical reasons. I'm not nostalgic about the game, I didn't play it anywhere close to when it was released.