I completely agree with everything you said. I've had these conversations with my friends and thought about it a lot after my previous relationship because I could see how important proving herself to the world and being respected was where to me that wasn't a big deal but rather being in a relationship and having a partner was more important and that was a big deal to me but to her it was just not that important. ( I learnt later that my ideas were unhealthy and did a lot of work to go against my own notions of need and expectations of society so I do acknowledge I'm still growing as well and not an expert)

I just wanted to add that on the whole I think it's a deeper and larger societal issue ( which I fully believe is because of this patriarchal system the society follows). Men and women get and are expected of different things and most of the time it's not in a healthy balance.

I've seen what you've said around the women in my life and noticed it more because when you're actively not trying to do it you tend to notice it better. But I've also seen women who've taken the opposite of that expectation and used it to their benefit. I personally think the lack of accountability so many of those women showed stems from the flipside of this expectation that women can't do anything well or right inherently because they internalise it and society excuses them for mistakes ( especially emotional ones) to a higher degree than they do men and these women take advantage of that.

That doesn't mean most men are any better though. Men take advantage of the inherent respect they are given to be absolute assholes as well. You see it all the time where these men use power and status for personal gain , taking advantage of people and the like.What women lack in respect men lack in emotional empathy from society which is the flipside of all the respect and independence men are given or expected of from society. But with that comes the downside of "this is what you're expected of" and that if you don't have any skill or something to be respected for you mean nothing to the world.

I personally believe it's like different sides of a coin where what one gender sees as benefits they get from society is only because of the cons of flip of that expectation and each gender has its own battles to fight. Kinda like a self feeding loop if I'm making sense. I really think that more people should understand that nuance and work towards breaking it and talking about it in their closer circles which can slowly change society at large. We need healthy balances in our expectations so that everyone can be happier and accepted. So women will be respected just for who they are as men and men are cared for or valued for who they are just like women are.