There's a common complaint about the Bioshock games, which is that they're not very good shooters. People writing about Bioshock Infinite tend to mention this, saying that the story is interesting and the writing is sharp but the actual game is poor. And this is true: it's not a very good first-person shooter, and it's arguably much worse than its predecessors. But this implication of most of these comments, from Kotaku's essay on its violence to Brainy Gamer's naming it the "apotheosis of FPS, is that Infinite is bad in many ways because it's a first-person shooter--that it's shackled to its point of view. In doing so, it has become a sort of stand-in for the whole genre, from Call of Duty to Halo.
I sympathize with the people who feel like the game's violence is incoherent (it is), and who are sick of the whole console-inspired manshooting genre. But I love shooters, and it bugs me a little to see them saddled with the burden of everything that's wrong with American media.
Set aside Infinite's themes and its apparent belief that the best superpower is the ability to literally generate plot holes--when we say that it's not a good FPS, what does that means? What is it, mechanically, that separates the two? I'm not a designer, but as a avid FPS player, there are basically three rules that Infinite breaks.
First of all, the enemy progression can't be just about "bigger lifebars." A good shooter increases difficulty by forcing players to change their patterns because they're not able to rely on the same rote strategy. Halo, for all its flaws, gets this right: few of its enemies are actually "tough," but each of them has a different method of avoiding damage, and a different weapon style. By throwing in different combinations, players are forced to change up their tactics for each encounter, or even at multiple points during the encounter. Almost all of Infinite's enemies, on the other hand, are the same walking tanks, with similar (dim-witted) behaviors and hitscan weaponry. I never had to change my approach, only the amount of ammo I used.
Along those lines, weapons need strengths and weaknesses. Each one should have a situation where they feel thrillingly powerful, as well as a larger set of situations where they're relatively useless. This doesn't have to conflict with a limited inventory--I loved Crysis 2's sniper rifle, spending the entire game sneaking between cover positions in stealth mode, but it was always paired with a strong close-in gun for when I was overrun. A good game forces you to change weapons for reasons other than "out of ammunition." Infinite's close-range weapons feel identical, and its sniper rifle is rarely useful, since a single shot alerts everyone to your position.
Finally, every fight cannot simply be about shooting. Most shooters are actually about navigating space and territory, and the shooting becomes a way of altering the priorities for movement. Do you take cover, or dodge in the open? Do you need more range, or need to close on an enemy? The original Bioshock made the interplay between the environment and your abilities one of its most compelling features: electrifying pools of water, setting fire to flammable objects, flinging scenery around with telekinesis. But at the very least, you need an objective from time to time with more complexity than "kill everything," both as a player and in terms of narrative.
Bioshock Infinite has, in all seriousness, no period I can remember when my objective was not reduced to "kill everything." Combined with a bland arsenal and blander enemies, this makes it a tedious game, but it also puts it at complete odds with its characters. The writing in Infinite is unusually good for a shooter, but it's hard not to notice that Elizabeth freaks out (rightfully) during one of Booker's murderous rampages, comes to a cheery acceptance with it a few minutes later, and then spends the rest of the game tossing helpful items to you under fire. That's writing that makes both the narrative and the mechanics worse, by drawing attention to the worst parts of both.
It's not the only shooter with those flaws--people just had higher expectations for it. The average FPS is badly written, and it's a favorite genre for warmongering propaganda pieces. But that's true of many games, and yet we don't see pieces talking about the "apotheosis of platformers," or talking about RTS as though they're emblematic of wider ills just because Starcraft II is kind of a mess. And there's still interesting stuff being done in the genre: Portal and Thirty Flights of Loving come to mind. To say that FPS have reached their limits, ironically, seems like a pretty limited perspective.