As I grew older, my parents explained to me that it boils down to two elements:

1. A home full of closed doors doesn't feel lively, and feels hostile in a way
2. Kids shutting doors mostly comes as they grow into adolescence, marking a shift in relationships. They get more closed off, this worries the parents, and the closed door is one visual representation of it. It looks like children lose trust in their parents, and closeness with them, and it's traumatic

Of course, restricting this does nothing to resolve the underlying issues. But parents, too, are emotional beings.

Because many parents are unreasonably frightened of the prospect of their children discovering they have genitals, and then {gasp!} touching them.