Women can not "choose" to remain covered anymore than a beaten wife can "choose" to stay with Their partner. It's not a choice, it's just a state of being. They might think they're happy with it but that's just because they're brainwashed
This is way too simplistic. Yes, some women may be forced to cover up. Yes, some women may be influenced by cultural norms to cover up. But some do choose to cover up of their own free will.
I used to live overseas in a non-Muslim country. I had a Muslim friend from a different country who wore Western dress when I first knew her. She subsequently made a decision to start wearing the hijab. Her husband was horrified and did his best to persuade her out of it, as he thought it would make it difficult for them to integrate properly in the country where they were living. She was very clear that she was going to stick with it, because that's what she wanted to do.
Obviously the burqa and the hijab are different, but we can't make assumptions about any of them. I can totally see why some women - regardless of religion actually- might prefer to wear a burkini at the beach. Banning them and telling women to wear fewer clothes cannot be a good thing in my view.