Stop beating the " Mercury 13" dead horse! Those women did NOT "participate in training" as you dishonestly state. They underwent SOME of the same physiological tests as the actual astronauts, but that's ALL. They never even met as a group until decades later!
NONE were military test pilots with experience in high-performance aircraft (one of the primary qualifications for the first few NASA astronaut groups); Jerrie Cobb, the most famous of the women, had never even flown a jet!
At the time of the first astronaut selection, no one had ever flown in space The first astronauts would be operating a new and untested vehicle in a completely novel environment. You can argue that the U.S. military was behind the social evolution in not having already utilized women and more minorities in the ranks of its aviation communities, but NASA's insistence on military test pilots was vital, not simply a "straw man" used to screen out all but white males.
I know there are some folks who love to moan "oh how unfair to my particular demographic", but the fact is that there were well-considered requirements for the early astronauts, and there weren't any women who met them.