If men and "not men" can ask for woman drivers they are "competing" for some drivers and making it harder for each other to get them. If only "not men" can ask for women it is easier for them to get the driver they want. So if men don't have a strong preference it's easier for other to get what they are asking for.

I'm not agreeing with them, just trying to make sense of it.

Maybe it's about men preferring female drivers and making it harder for other to get them. Woman may request a female driver to feel safer but men provably don't do it so much for that reason.