I believe that many years ago, people put large leaves on their heads to shield against rain and sun and wind. Eventually, they put pieces of cloth on their heads for much the same reason and saved the leaves to use to wrap fish in.
Then, God said Hi to Abraham and brought him a casserole. No fish.
Many years later, while people were still using the cloth to keep out the hot sun or the rain or the sand or the mice, God said to a guy, "as part of my code, I need you to show subservience, humility and respect, even if you are inside or there is no sun, rain or wind."
That guy had a full head of hair and a long beard (and mustache...don't get any ideas) but he also had respect for the law God gave him so he talked it over with his friends and they all agreed. They said "we cover our heads to show that we are less powerful than the elements, but God is more powerful than the elements, so we should CONTINUE to cover our heads even when the elements aren't around, because God is!"
"Brilliant!" many of them shouted except for a small group who said "maybe just when we are actively mentioning and praying to God" so the two groups went their separate ways, with 2 separate understandings of when to cover their heads vis-a-vis God.
So, no, it had nothing to do with fashion, or baldness. Or mice.