Do you speak French? I am on a lot of expat FB groups and so many questions people have are down to not speaking French. Those with holiday lets have mixed experiences. Some do OK, but a lot struggle to get bookings, plus have to deal with some proper nightmares caused by guests.
Your children would learn the language and adapt quite quickly. It is often the parents who have more trouble. University is a lot cheaper so if you are within commuting distance of one of the good ones I wouldn't worry about them being disadvantaged. The education system is, from what I can tell, easier to get to grips with the younger they start in it.
What would you do to fit in? The happiest people tend to be, from what I can see, people who have made an effort to learn the language, accept France with its flaws and accept that things are done differently there. Red tape is a bit of a shock for people coming from the UK, but that is just the way it is. People who expect things to be as easy in the UK as far as business stuff is concerned get a shock. Same with food, shops, services etc - it is different in France. Getting things done in August, in fact at any time of the year, might be a challenge. Shops aren't open all the time. Customer service is different (I don't think it is bad, it is just very different).
The culture shock is far higher than people expect it to be, and people get very lonely very quickly if they don't accept that things are done differently, or just mix with other expats.
I think it is a fabulous country. I have lived here most of my adult life, have married and had children here and have my own business here. It's my home. But I think some people have had quite a hard time of it.