I really have no idea what this side tangent is about.
I made a comment that getting ready twice in short order is a pain, which is why people wouldn't want to have PE first thing.
I don't see what any of this has to do with the difference in showering schedules between white collar and blue collar labor.
My dad worked in a factory and showered after he got home from work. That was the time of day that made sense because his work was hot and sweaty and dirty.
I suppose the idea could have been that you could have just not showered at all and showered once you were done with PE, but still unsure where this detour is headed.
It’s a class marker, a sign of social status. Sweating is shameful, as is being low social status. You don’t see your father’s job, which he showered afterwards, as something shameful but loads of middle class people would because they’re insecure in their social status, they associated with people of similar income but lower social status. People with much higher status don’t care because they don’t, there’s no chance of them being associated with those with lower status.
You shower after manual labor jobs. That's just a matter of practicality.