As you said, it's mostly a joke, usually exaggerated for comic effect. Even calling it "anti-French" may be too much; it's just playing off a stereotype of French people as snobby and overly "sophisticated" (in a slightly derogatory way). I don't want to link it to working class/upper class or rural/urban or whatever, since those aren't really right, but perhaps a vaguely similar impulse.

Trying to tie it to "foreign policy" or something is thinking about it much too hard.

That stereotype exists because, right before US elites started formalizing their own cultural scenario (before F.S. Fitzgerald, to give a rough idea), French culture was one of their main models, if not THE model. On the other side, French elites invested a lot in the American Revolution and were quite disappointed with the outcomes (a society characterized by freedom but also rough and greedy, which eventually chose The Anglo Way in most matters and stole their thunder as Beacon of Civilization). Ever since, both sides were locked in a constant and intense debate in a way that very few other countries experienced. It's funng because both French and US mainstream cultures agree on very fundamental matters (the need for a secular State, the power of technological progress, the importance of cinema and popular culture, down to comicbooks) but express them in completely opposite ways in practice.

There's a very complicated history of mutual love and disappointment between France and the US, stretching all the way from before the American Revolution.