Well, I think people "naturally" tend to cover their genitals for a variety of reasons which we need not concern ourselves with. But beyond that, what do you need to cover is mostly cultural and I think that in this respect culture is mostly driven by religion.

For example, most pre-religious people do not care about women going topless. But Christianity is pretty sure women should cover their breasts. Traditional Judaism goes further and says that married women should also keep their head covered at all times, that's why married Jewish Orthodox women wear wigs. Islam agrees that hair should be covered but in many places goes further and says most of the face should be hidden as well.

In, say, contemporary Christianity-based American culture women can't normally go topless -- that would be offensive to many people. But a hundred years ago a woman in bikini would also have been offensive. And a woman with uncovered head and open face would be offensive to some Muslims.

if they were inventing a ban, they could have just said it is embarrassing because it is bad.

I think religion is more sophisticated than that :-)