I would indeed define successful integration as you know, "behaving French", adopting French customs, speaking French at home, etc.
Why is this desirable?
Depends on what you believe. Some say a national identity is important in order to have social cohesion, a narrative that unites all the different segments of the population in a common ground they have in order to favor of the country interests and a sense of peace. Otherwise people will fight themselves.
Every country has their own national identity, otherwise what really is a country other than a random piece of land that has a bunch of people with nothing in common that hate each other, having to live with one another?
Problem with France is not that it’s trying to protect their national identity, but their misguided strategy in order to achieve that goal. But other countries are also hypocrites in their construction of different internal narratives and identities, so no one can really judge France too hard imo.
France only gets that much attention because their national identity have strong particularities and are very different than the one constructed in America for example.