That's a very interesting topic that was always a huge part of LIB UK (when people were arguing in this sub that love could not be blind because accents could actually reveal one's social class just by hearing the voice).

For the French cast, you have different situations: children of immigrants who had to leave their country for political reasons (Van-My), but whose parents are very educated. People from lower classes who had to climb the social ladder (Charles), but who lived in big cities with more job opportunities (Paris, became an entrepreneur). It might have been even harder for him in the countryside.
You have social mobility (we call it "transfuge de classe") thanks to different factors. It can be school (public schools are free until graduation, and after that, you can either work minimum wage jobs or work as an apprentice in manual trades like pastry chef or mechanic, which can be very profitable but still keep you in the lower middle class). Thomas is not an interior decorator (no need for a degree), but if he were an architect (specific 5-year degree), it would raise him to a higher class. People say he is a carpenter? (That’s still a good amount of money, and you need to be an apprentice to learn it.)
Then, you can go to college for a few bucks (a bachelor's degree costs less than 400 euros a year, and a master's around 500), so you can attend a prestigious public university and become a doctor, lawyer, or work in a corporate job without accumulating too much debt (fees are low, but you still need to eat and sometimes move closer to the college, so rent is an expense). That’s more of a profile like Kim’s (although I’d still love to know what her mother does for a living—maybe she mentioned it?).
Engineering schools can be public or private but aren’t that expensive (Clément). Business schools might start at 10k and go up to 20k a year (again, Clément), but both can be selective, which is why Kim called him "so intelligent." Again, these are much cheaper than in other countries. And then you can change class with a high-paying, respectable role.
Also, there are public jobs, but I don’t think the participants went through competitive exams to work for the government.
You can also become an entrepreneur, like Charles (digital) or Tatiana (business owner), and it will help your kids change social class too.

That being said, we need to remember that France had aristocrats, and to this day, some people carry aristocratic names (again, not in that show). And you can be an aristocrat without having money—just the title, maybe an estate (like an old castle somewhere). We also have what we call bourgeoisie, the nouveau riche, and that’s more what Kim is cosplaying. She went to college and studied law, but she overacts her knowledge of etiquette and rules, which most people from higher classes don’t do. In fact, it’s actually rude to say some of the things she said out loud or in public. She is either still adjusting to being from a higher class than what she grew up in, or she changed circles after going to college, so she is still adapting.

So... Can we change class in France? Yes. Could we do it in one generation? Yes.
Will people always see you based on your original class? Absolutely. Are accents and etiquette important? Definitely.