Can you point to examples in Europe - apart from some Muslim-majority countries - where women are legally or socially required to cover their faces or hair?

Head coverings aren't the only form of restricting bodily autonomy. I'm talking about abortion. Something which religious people, Muslim and Christian, push to actively ban.

Christianity went through a historical process of separating religious belief from many aspects of civil law and political authority

Christianity did not go through this process. The people under Christianity went through the process of enforcing separation of religious belief and civil law. There are people in Ireland who are still alive and remember when the catholic Irish constitution stated explicitly that a woman's role is in the house. Laws were created around this, including limiting the job opportunities of married women. The movement to lift this was pushed by all women, not just Christian women. Morocco also had laws restricting womens right to work, and women, including Muslim women, pushed to lift them.

I see Islam as more than simply a religion in the conventional sense: it is also a comprehensive legal and social doctrine

All religions are comprehensive with their social doctrine. A key aspect of religion is telling people how to live and why. Different groups at different times have chosen whether or not to enforce their own lighter or harsher interpretations or misinterpretations of these doctrines on the people.

My point is Mussolina did not ban crucifixes.

Nuns get sent to convents, historically not by choice.

It's just two sides of the same coin.

Hijab is also a personal choice