As others have said, the strength of this nation is the diversity of its people's heritage. You look at the US Olympic team or you look at their names and you can see almost every corner of the world represented in some way. USA is top of the medals table because of these people and the people have their medals in large part because they competed with USA on their jersey or singlet or swim cap or leotard. One without the other is one-handed clapping.
I'm an immigrant, in fact unlike Leo Manzano I'm a LEGAL immigrant. I came to the US at the age of 25 so even if I had been a top athlete the question never would have come up, but as a thought experiment, given the choice today I would represent my home country in the Olympics, no doubt. I assume Manzano had that choice but he opted to represent the US and that's where he made his bed. If he wanted to wave the Mexican flag he should have run for Mexico.
To keep it personally relevant I'll fast-forward 15-20 years. I have a young son. He's born in the US and his mother is American. If he becomes a great athlete and has the chance to go to the Olympics the natural choice for him will be to represent the US. He's a dual citizen but if he should dare to wave the flag of MY country I would be embarrassed, not proud.
However, if he's not good enough to make the US team in whatever sport it is and he has the option of going to the Olympics as part of another team, I sincerely hopes he does that. Would be stupid not to.