And how many American kids today would carelessly shrug if you asked them what they thought of the US?
In some modern day military recruitment drives, you'll hear things like "America *is* the Army" or t"he Marines Make America." It's not unusual to combine patriotism with service in order to further a point to an otherwise apathetic youth.
This movie's Kirk is rebellious, crashing cars, flirting with chicks, getting in fights, and more intelligent than he initially looks like. He's anything BUT apathetic.
And "Federation is Starfleet", "Starfleet makes the Federation" are certainly not slogans you'd expect to hear in the Star Trek universe. Starfleet is exploration first, defense second, war last.
No, Starfleet is exploration and defense co-equally. And do bear in mind that Pike specifically cited
peacekeeping and
humanitarianism in his description of Starfleet -- not exactly the same as saying that Starfleet is all about kicking some Klingon and Romulan ass.
And, yeah, "the Federation is Starfleet" is not something you'd normally hear, but obviously Pike has a particular POV that he was imparting to Kirk in an attempt to get him to join Starfleet. Not everyone in the future, even a utopian future, is going to be politically correct in their opinions. It's really not a big deal.
Case in point, a specific tactic by military recruitment is to target kids and young people who otherwise have no discernible or chosen future but otherwise have no real patriotic allegiance. Or many recruits join not out of love of country first but for the promise of ensured college education. This Kirk seemed to fall into both camps.
How could you say that about nuKirk after seeing him ponder his father's fate (when holding the
Kelvin salt shaker) and then pondering his own potential (when looking at the
Enterprise under construction)? Kirk rather obviously joined Starfleet because he realized it was a way to improve himself, to become a better person and a good leader. To command a starship.
In other words -- the same reasons Prime Kirk joined Starfleet.