It is true that in a purely rational system, there would be no pay gap (controlling for ability and impact).
But guess what...
But guess what...
They care about one thing, dollars. If you are making them money, they will keep you no matter what. If you are not making them money, they will get rid of you. Pretty rational I think.
Corporations are run by people, and they draw on a huge variety of sources to make their decisions. Including, obviously, some that are not rational.
If women choose to forgo higher salaries in favor of something else, that's a different issue and it doesn't immediately mean that they are being "irrational", it just means they favor different things than men do.
If the hiring managers (of any gender) have an irrational bias towards thinking that women are not as effective as their male counterparts, this will affect the respective pay scales even within a competitive market.
Note that I'm not providing evidence or making a claim about the gender pay gap. I'm just saying that the claims you've made have many possible consistent explanations, including irrational bias against women.