One of Halo's major innovations was that it removed the health bar. In a shooter like Doom or Goldeneye, you have let's say 100 HP, and different weapons deal a different amount of damage. It's just like any fighting game with a health bar. Halo replaced that with a shield system, wherein being shot repeatedly wasn't a game-ender if you could get away long enough for shields to recharge. It changes the pacing of the game substantially, especially in multiplayer. The jittery feeling of older games, where every shot counts and you always have to be on the lookout for your opponent, was made both slower (because other players weren't constantly trying to get a shot on you and wear you down) and a bit more mindless (because once you got into a firefight, your only choice was to shoot the other person repeatedly or else there'd be no damage left whatsoever).
This complements the self-seriousness of the Halo games, which absolutely inspired the direction of the next-gen titles. Halo was "grittier" and less silly than previous shooters had been. In Wolfenstein, you were fighting enemies like robo-Hitler. In Doom, the enemies you were up against were inventive, silly, and fun. And the push towards realism was by no means inevitable. Look at Unreal Tournament 2004, which was a touted push towards enhanced graphics and gameplay that took advantage of modern systems, but which remains an utterly comical game. The violence is whimsical and comic; the levels are far more ridiculous than Halo's or Halo 2's. It's a pretty game, as far as that goes, but the prettiness is used for exaggeration rather than for "grit". So obviously that path was not only possible for games, but it led to quite a few very good games over the last decade.