Since he/she was born here, your hypothetical Disneyland trip baby has American citizenship. Whether it has French citizenship as well is an issue for French law to decide. The baby, when he/she reaches majority can renounce his/her American citizenship if he/she wishes to by going to an American embassy or consulate, filling out various forms, sitting though some required interviews to make sure he/she understands what he/she is doing, and surrendering his/her American passport, if one was ever issued to him/her. There are other bells and whistles on the situation but this is already long enough.
Should American law be more restrictive? I certainly think so. I am undecided as to how exactly to define the restrictions.
Only true if said parents entered through legal channels.