This is an instance where grammar, logic, and cultural ethos enter the wrestling ring; logic hollers uncle*, and whimpy grammar pleads "no contest" and meekly accommodates the victor.
'can' means 'is permitted to' in the sense that barriers of class distinction, race, and religion do not stand in the way. (The assumption here is that the child was American (USA) by birth.)
'could' suggests 'possibility', and to non-Americans such as you and I, seems the more logical, grammatically correct choice when we think of the odds. Such 'tentativeness', in a land where "nothing is impossible" is counter to the sentiment being expressed in "can be president", which overrides the logic of grammar: this is the land of the free, where anyone with ambition, drive, and savvy can achieve their dreams. There is no room for iffy tentativeness.
(Kennedy said they would, and they did put a man on the moon!)
Perhaps an American who lives and breathes the ethos can better express this sentiment, straight from a heart and soul that is imbued with the spirit.
* an American colloquialism, where to say or cry "uncle" is to indicate surrender or admit defeat.
holler : colloquial - "give a loud shout or cry"