France is an officially secular state, with a comittment to sex equality.
The Burkini makes a religious statement, and like the burqa and the hijab, has connotations about women's inferiority that are incompatible with French values.
Go on a French beach in a wetsuit - nobody minds. If an Indian woman goes on a beach in a full sari as they do in India (indeed men in full suits can be seen on Indian beaches) - no issue as it's saying nothing about the position of women, nor is it an expression of religion.
You will see women in France flouting the face covering ban, and yes, I think they should be challenged. I don't want to have any interaction, personal, social or professional, with somebody I can't see. And it is a security risk - banks have been held up by burqa wearing men.
I don't buy the idea that the world's burqa wearing women are free thinking feminists who choose to wear a curtain. Some wear by choice - some because of pressure or fear, and that bothers me.
I have an entire family-in-law who are muslim, and none of the women are veiled. It is neither necessary nor "Islamic" per se. It's a misue of religion to oppress women.
Incidentally, I would equally say get the veils off those nuns - an expression of Christian extremism.