Let's set aside the burqa since it is such an extreme example. Let's talk about the hijab instead, which is much more common, and less objectionable.
I don't see why non-Muslims need to be freaked out by seeing the hijab, any more than non-Hindus should be freaked out by seeing a tika, or non-Christians should be freaked out by seeing a cross pendant.
As for respecting women's, or anyone's, right to be less religious than oneself, then I will agree that many Muslims need to become more tolerant of nonobservant Muslims.
Now, coming to your central point about expressions of religion in public, I will give the contrasting examples of American and French style secularism. The French way is, as you mentioned, to force people to hide their religion so as no one is "forced" to look at others' religion. The American way is the opposite: it allows people to express their religion but requires them to respect others' right to have their religion.
The American way is more fair since it recognizes that the majority will always find some way to showcase its religion and create exemptions for itself, and that laws to hide religion will penalize minorities disproportionately.
So, the better approach is not to hide the differences, to deny the diversity, but to celebrate it by inculcating tolerance and mutual respect.