Man, this "WHY IS BIOSHOCK INFINITE A SHOOTER" thing irritates me. BioShock Infinite HAS to be a shooter. That's the best way to convey the setting. Columbia is a dangerous, dark, violent place. If the entire game was just exploring the city without incident, that would not be true to the story. Comstock is a dangerous, violent man. Booker DeWitt is a dangerous, violent man who has committed horrific atrocities in his past and this is his last ditch effort to save himself. There is no other way for anybody or anywhere in the game to be.
I disagree with your premise that only shooters can convey danger and violence. I disagree with the implication that conveying danger and violence are BioShock Infinite's primary goal. I especially disagree with the false dichotomy that the game can only be either a shooter or "exploring the city without incident". (And for that matter, "exploring the city without incident" would be true to the story, it would just be true to a different story.)
I don't have a problem with the fact that BioShock Infinite is a shooter per se; my issue is that it is most of the time a mediocre, mostly-linear shooter, albeit one with outstanding art direction. It adds a couple of gimmicky mechanics that seem cool at first glance but add little depth. And, as BioShock nodded toward objectivism without having a great deal to say about it, BioShock Infinite invokes American exceptionalism and racism only to say very little of substance, and in a ham-fisted way at that.
For all that, I don't hate BioShock Infinite. I think it's okay. But the first hour or so, when it seemed to be trying for something more, that was the part of it I most enjoyed.