In the case of Halo, I think they hold up remarkably well. They were designed with a very different sensibility than most single player FPS games today. Halo encourages the player to jump right into the hordes of enemies and take them on in a very aggressive, but often strategic, assault. It's rarely about sitting back and taking pot shots. It's a much more dynamic type of shooter almost in the style of an old school Japanese style arcade game or something.
Also like an old school game each type of enemy has a different sort of weakness which encourages threat prioritization from the player.
Finally the way they mix up enemy combinations and formations makes for some really interesting battles. It's really the only reason why Halo CE's repetitive levels actually work - enemy placements and encounters are excellent regardless of scenery. Of course, level design can still work in tandem with this to create some interesting arenas.
Some modern games still take this approach, thankfully, but few have mastered it quite like Bungie. All of the systems they designed for each Halo game just work so damn well together.
That's part of where I think H4 falls down - the encounters just aren't as interesting and often go on too long. Worse, you're constantly running out of ammo for preferred weapons and are required to change up all the time. If you played well in Bungie Halo games you would generally have enough ammo and grenades handy for preferred weapons when you needed them along with something in the back pocket. Halo 4 just doesn't have the fine tuning or balance that Bungie's games did (plus there are too many arena-like areas where you're stuck fighting loads of enemies in one space).
A lot of other FPS games have aged badly, though. I don't really think the Half-Life series holds up all that well today, for instance, despite being so remarkable at the time of release.