Yeah, Bioshock: Infinite had problems. So many problems that I could go on for a very long time about everything that was wrong with the game and its story. But, I won't do that. I'll just go on for a little while and refer you to smudboy's excellent 5 part analysis of the game's nonsensical plot.
First off two weapon restriction completely ruined the gameplay. Okay so the two weapons limit is supposed to encourage me to switch up weapons, but why then is there an upgrade system for weapons? As soon as I pour any money into a weapon I then have to stick with the weapon forever because otherwise there was no point in upgrading it at all. So basically I played most of the game with the same two weapons (carbine and shotgun), never really getting a chance to play around with the other weapons because I was scared that if I dropped one of my guns to try out a different one I wouldn't be able to find another one of my upgraded guns when I needed it.
Second, the people were so static and really didn't fit with the story that the game was trying to tell. For about a quarter of the game I was under the impression that everybody in Columbia was insane due to the way that they were built up in a similar fashion to the splicers in Bioshock, what with all the dead mutilated bodies and the fact that all these guys charge at Booker with so much gusto when he is clearly a walking genocide. Around the time that I got to some shipping yard and saw all the regular people standing around and cowering I started to suspect that I was actually supposed to sympathise with these cardboard cutout people and wasn't supposed to have concluded that the entire city was an insane cult or something.
Another thing that really bothered me was the whole deal of trying to get that stupid airship back. It was nonsensical, even disregarding all of the retarded alternate dimension stupidity. Booker blows up like 5 more airships, why can't we take one of those? Seriously, we just hijacked that first airship, why did Booker immediately form such an intense bond with the first blimp he steps foot on?
Lastly, why do we have to stop Comstock from existing in all realities? Seriously, there are infinite dimensions, why can't those other Bookers and Elizabeths just look after their own crap? And it's just one tyrant, there have been thousands throughout history, multiply that by infinity for every possible tyrant in the multiverse. What's next, a quest to stop every single person who has ever done a bad thing in any possible reality across all planets over the entire history of time? Because that's only slightly more unrealistic that deciding that it's your job to stop every instance of Comstock over an infinite number of realities. It's like if I see a starving person and I go and feed them, and then I decide that I now have to go and feed every single hungry person that ever existed, exists, or will exist. Like holy cow, you don't have to personally take responsibility to fight every evil there ever was, there's only so much one person can do.
So now that I've gotten that out I want to refer everyone to watch smudboy's analysis for a great in depth dissection of everything that was wrong with Bioshock: Infinite's story.
Seriously, if you want in-depth analysis of Bioshock: Infinite, here it is. This is about the most depth you will find on the subject, 90 minutes of it to be approximate.
One last thing, I find it kind of funny how many people there are in this thread saying how bad Bioshock: Infinite was when at launch it was pretty much lauded across the board. I guess most of the cynical critical people don't buy games at launch, which would make sense.