That cross-dressing tradition comes from the very opposite direction you’re assuming.

For example in their time Shakespeare’s Plays were entirely played by men, including female roles. This was not because they were very forward thinking about cross-dressing back in 18th century England, it was because it was deemed inappropriate for women to be actors.

It’s a reflection of the so called “Benevolent” Sexism still widely prevalent in those isles, especially in the Middle Class and above - women “are emotional and fragile creatures” and “must be protected” (were do you think the phrase “the gentler sex” comes from?), a paternalist take that justifies denying women access to certain things “to protect them” and certain responsibilities “because they’re too emotional and unlike men can’t handle the stress of such posts”.

You’re seeing the visible, surface artifacts of a society and applying your own values (very likely those of somebody with a high level of formal education and a big city upbringing in one of the most forward thinking areas in America) to try and derive the reason for those things when what you’re looking at is a society which is has been highly calcified socially for centuries, which, except for a bright period after the War that lasted until the 70s, has been dominated in terms of Power and Culturally by a very small section of society whose privileges are passed from generation to generation and were appearances are far more important than action, so with values massively mistatching yours, hence you’re drawing erroneous conclusions.

It’s not cultural openness that drives them, it’s various alive and well streaks of Victorian Values mixed with an unhealthy concern with “keeping up with appearances” (curiously there is a humoristic TV series with that name and tellingly it makes fun of Working Class or Low Middle Class people some whom desperately trying to look higher class, though probably one needs to have lived in that country for a couple of years to read the social context)