So your theory is [...]

Nope. (What did I say to make you think that?)

My position is as follows.

• There may well be differences in average performance between men and women in various intellectual tasks, in either direction. Indeed, there are some specific categories of tasks where the evidence for such differences seems strong; the most famous example is probably "mental rotation". The difference for mental rotation is large but not enormous (about 1sd); my understanding is that all other sex differences found in scientific studies are smaller, and there are differences going in both directions.
• It seems unlikely to me that there's a big general cognitive deficit on either side. I believe girls are currently doing better than boys in pretty much all subjects at school in my country nowadays; in the past it was the other way around; so whatever differences there are (in this kind of task) must be smaller than the size of difference that can be induced by cultural effects. Of course this is consistent with deficits in very specific areas, with variance differences that affect how many really stellar performers there are of each sex, etc.
• There may well be differences in variance between men and women. These differences might be fairly big, but it seems unlikely to me that they're large enough to make huge differences at "ordinary" ability levels.
• Once you start looking at the tails of the distributions, I expect them to be quite far from being Gaussian or even symmetrical. There are a lot more ways for things to go badly wrong than for them to go exceptionally right, after all. So I am skeptical about inferences from differences at the "low" end to differences at the "high" end.
• There are certainly a lot more men than women in prison, especially if you look specifically at crimes of violence. However, lumping this together with variations in ability seems like a wilful embracing of the halo/horns effect; it seems like much of it will come from variations in sociopathy, enjoyment of violence, and other such characteristics that needn't go along with worse performance as (say) a scientific lab manager.
• For those last two reasons, any inference that looks much like "there are more men in prison, so we should expect more men to win Nobel prizes in physics" seems extremely suspect to me. Still more "there are more men in prison, so we should expect more men to make good lab managers".
• Putting all the above together: there may be well be differences in competence between men and women; they may well be bigger at the highest levels; I wouldn't expect the differences to be enormous except maybe at the very highest levels where variance differences can be a really big deal.
• All of that is independent from the question of whether there are sexist attitudes -- by which, for present purpose, I mean: whether there are systematic biases that make men get evaluated better than women relative to their actual ability, likely performance, etc. Or, for that matter, worse.
• It seems to me that there is a lot of evidence that there are such sexist attitudes, generally favouring men over women. We've had a lot of discussion of one study which seems to me like very strong evidence for such attitudes in one domain; I posted some links to some others. There's a pile of anecdote too, but of course the way that looks may simply reflect what anecdotes I happen to have encountered. (I think the available anecdotage is at any rate evidence that sexist attitudes in both directions exist.)
• The possibility of real ability differences has some bearing on how to interpret the apparent evidence for sexist attitudes, but in at least some cases -- e.g., the study we've discussed so much here -- it seems to me that it doesn't make much difference, because to make the evidence not be strong evidence for sexist attitudes it would be necessary for the ability differences to be (what seems to me to be) unrealistically large.
• The relevant question for most practical purposes is not the statistical difference between men and women in some particular kind of ability, but the statistical difference conditional on the information usually available when hiring (or when considering promotion, or when allocating places at a university, or whatever). Nothing I have seen so far gives me reason to think that these differences are large, even though the information in question is limited and unreliable.

(To get quantitative for a moment, let's suppose everything in sight is normally distributed. Some underlying ability: male and female both have mean 0, but s.d. 1 for men and 0 for women. Some measure of ability equals the actual ability level plus noise with mean 0 and s.d. 1. Actual job performance looks like underlying ability plus other factors with mean 0 and s.d. 0.5. Then conditional on measured ability being +2 (i.e., well above average but not stratospheric), mean predicted job performance is about +1.0 for male applicants (with s.d. 0.87) and +0.8 for female applicants (with s.d. 0.80), a difference of about 0.2 (male) standard deviations. Definitely not zero, but not exactly huge either and a lot smaller than the noise. I have no idea how realistic any of the numbers I've assumed here actually are, and would be glad to learn of credible estimates -- though of course this is a toy model at best whatever numbers one plugs in.)

If you think that an argument is good just because you think its conclusion is true [...]

I agree, that's a very bad sign. On the other hand, there's nothing very alarming about thinking an argument is more persuasive when you agree with its premises. And often the premises and the conclusions are related to one another. That seems to me to be exactly the situation here.

Premise: There pretty clearly isn't an enormous cognitive difference between men and women that makes women much less competent at brainwork, so much less competent that a moderate amount of information about a person's abilities leaves a lot of male-female difference un-screened-off.

Argument: If indeed there isn't, then the best explanation of findings of the sort we've been discussing is prejudice in favour of men and against women that has substantial impact on hiring.

Conclusion: There probably is such prejudice, and it probably leads (among other things) to underrepresentation of women in many brainwork-heavy jobs.

(Note: the premise, the argument, and the conclusion are all sketchy approximations. Filling in all the details would make the above maybe 20x longer than it is.)

I find the argument somewhat persuasive. This is partly because I find the premise plausible; some people might not (e.g., because the evidence they think they have regarding the relative abilities of men and women differs from the evidence I think I have); those people will find it less persuasive.

The premise in question is not the conclusion of the argument. It is not equivalent to the conclusion of the argument. It neither implies nor is implied by the conclusion of the argument. It is, for sure, somewhat related to the conclusion -- e.g., by the fact that they are premise and conclusion of a short and simple argument -- and doubtless there is a correlation between believing one and believing the other. I do not find this sufficient reason to think that finding the argument more credible if one accepts the premise is any sort of cognitive error.

Perhaps I am misunderstanding your argument somehow. I confess I don't find it perfectly clear. Would you like to make it more explicit what error you think I am committing and why you think that?

Even if men and women are on average equally qualified that doesn't mean that a specific subset is.

I agree and am not aware of having said or implied otherwise. One of the many modifications that would be needed to turn the argument-sketch above into something unambiguous and quantitative would be to replace "between men and women" with "between men applying for lab manager posts and women applying for lab manager posts". If you think this makes an actual difference in here, I'd be interested to see the details.

For a hiring manager it's not important whether there's causation. Correlation in the data set is enough.

Depends on the details, of course, but mostly yes. Once again, though, I am having trouble working out what I've said that suggests I think otherwise.