A contradiction in the liberal view of gender

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That's really true. It's what bothers me about some affirmative action things like 'Women in STEM' awards. If you put false incentives on something (like women pursuing STEM) you're not going to have a natural proportion of people pursuing it. I think it can be different if one group is clearly disadvantaged, but for something like science it's not like girls have fewer opportunities to study it than guys (at least in Canada, I'm sure its different elsewhere).

-and the other foot finally falls.  Something else bothers you.  

Plenty bothers me

I don't think that's an accurate parallel to draw. Adopting one of two extreme gender roles does not bridge the gap between them, as bringing food to those who don't have it bridges the gap between those who have food and those who don't. One is actively solving the problem, one is more of a compromise.

Who's looking to bridge any gap in either statement?  What gap?  What compromise is being made in either statement?

'Changing genders' implies that there are distinct genders and you are choosing one of them. Like, if I change my job, I distinctly no longer work in one job and I distinctly work in another. If you don't believe that genders are distinct, sure, choosing between two is better than being forced into one, but the full extent of your view is that you shouldn't have to choose between two.

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No, they believe that they are arbitrary.  You're the one that chose to take the statement that they were arbitrary and then launch into something about them being invalid.

Well, sorry if I didn't make this clear in the OP, but that is the ideology I'm referring to.