The handful of Polytechnicians I’ve worked with (I’m on the board of a French company, although it’s owned by a US PE firm) are very impressive (certainly at the level of Oxbridge, and academically more capable than your average Ivy graduate) and deeply connected within the French establishment (which means a lot when it comes to prestigious jobs in France). In my experience there’s a somewhat similar level of intellectual snobbery about X in France to Oxbridge in the U.K. (though French people are not at all self-deprecating like the English), my best description in US terms would be MIT levels of ability with Harvard levels of attitude.
Of course those connections might not be valuable or even accessible to an American student unless they plan to remain in France. And in a sink or swim environment, it will be extremely hard to keep up, especially as a non-native speaker.