you get armies and corps form Culture not science. Also some of the policy cards from Culture are very strong and have really cost whatsoever while most of the science stuff you have to build such as the buildings.
With flight tile improvement that give Culture also start giving tourism so civs that can build Culture tile improvements do have an advantage here.
The river requirement is a problem but that is their only requirement so rivers will be lined with chateaus.
Corps and Armies are really late to come in hand, even if you are fast in culture. And in my knowledge, corps and armies need mutiple units to form. I dont think they are really practical for france, when a grande imperiale needs 8-9 turns to produce. And you need 27 turns to make an army, while in those sweet 27 turns science focus civs can prodcue infantry to smash Grande Imperiale to the ground.
When talking about the advance of civic tree, China, Egypt, Greece, Rome and Japan are more powerful than france. The chateau comes at humanism, which is really late in the civic tree. Rome can spam cities with free monuments, Greece can have a higher culture output theatre district and Japan has extra adjacency bonus. Egypt and China are fierce early wonder builders. France is certainly shadowed in front of them. Except, if you are only playing against USA, then France will certainly win ( in terms of civic tree advancing). And in my experience, France will only barely catch up in civics after Chateau has come, but never get too much ahead.
Yes, they will give tourism after flight, if you survive beyond that. Even after you start to output tourism with the chateau, say you have got 10( which is very unlikely in normal game play, most of the time we can get only 6 chateau adj to wonders, if we even get wonders). You will get 40 extra tourism, which is a bit of seasoning, but not the main course of tourism. And if you are really lucky to survive to very late game, I am afriad that space victory may still be faster than those 80( doubled by computer) tourism advantage. But when France's strength is really not solid, I can only pray for a cultural victory.
Something may look nice on paper, but they just suck at practical conditions.