You're making some really good points here, but you're missing a few things.
To start out with, TRP's philosophy is based on the principle that women are inherently bad people. They don't want committed relationships, they're greedy, and they aren't worthy of actual emotional connection. This is different from the sexual revolution with women, which taught them to define themselves independently from men, not to tear men down in order to bring themselves up. Her son might very well be buying into the second philosophy, but TRP as a whole promotes the first.
One of the most important things to consider is the words "taken seriously". There are two very different perspectives, broadly speaking, that these words often represent. You don't take a relationship seriously when its success and future is not a driving force in our life, or you don't take a relationship seriously when you cheat, manipulate, or otherwise do immoral things because "it doesn't really matter". The first position is a very healthy one, the second one is very unhealthy. Again, TRP promotes the bad one- women are not equal partners or worthy of respect, they're just the end goals that should be manipulated and controlled. And, again, her son might be taking this the healthy way.
The last thing that worries me is the idea of a woman needing to "prove herself". This is accurate, but it leaves out the fact that both partners have to prove themselves. By verbally expressing only one half of the burden, it suggest that he doesn't take his own burden as seriously. In other words, the woman has to work to be worthy of his affections, but his worthiness stems from something intrinsic that doesn't require effort. Obviously this is problematic, but I'm also basing this on a secondhand and very brief account, so maybe he does see both halves of the burden. But, based entirely on the woman needing to prove herself worthy of being taken seriously... yeah, that's kinda worrying. It's worrying for a person of any gender to think that only one partner needs to actively work at self-improvement for a relationship to succeed because it suggests a highly skewed power dynamic.
There are definitely comparisons between women's lib and modern men's rights movements. However, TRP in specific promotes a philosophy that, while rooted in sound ideas of self-improvement and solutions to inequalities that screw men over, is also sexist and promotes conflict rather than communication -based solutions. I would be equally worried if OP had a daughter that was quoting sexist garbage even if it had a strong current of women's empowerment.
Long story short, there are serious problems being addressed but TRP offers unhealthy (conflict-based) solutions instead of healthy (communication-based) solutions.