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Not questioning what you're saying, just adding that I think "prejudice" is a better word for this than bigotry? (to be clear: we're discussing here the results of identity-based hiring practices, as opposed to are there sex differences in programming ability)

Words never have entirely cleanly defined meanings, but broadly I think bigotry is often used to speak specifically of all-out irrational dogmatic beliefs. Prejudice is more often used where there is some partly rational judgement about a group of people, together with moral problems caused by applying that logic to a particular person. Of course, often our prejudices are very fallible: "rational" prejudices turn out to be wrong, and in that sense are functionally equivalent to bigotry. But holding a "rational" belief that all prejudice is irrational also does not make us infallible!

Some prejudice seems hard to criticize morally: for example, everybody makes prejudiced judgements say based partly on clothing, age, and sex if they find themselves in close proximity to a group of young men in a city at night. On the other hand, at work, one tries hard to not judge based on whatever preconceived group notions one has -- I think almost everybody thinks that's a good thing (which as you say can be harmed by identity-based hiring). I don't have a good abstract explanation of what makes the difference between "good" and "bad" prejudice, and I wish I did, so would love to hear of good writing about it if somebody can recommend some!

In the company I worked for recently, perhaps even a majority of the more capable programmers around me happened to be women. But wherever we do start hiring based on identity, it's hard to see how prejudice can be avoided, even if bigotry were entirely absent.