Which means that we haven't been hiring the best and the brightest. We've been hiring those who are similar to us.
I agree, and I totally support blinding of resumes, and as much blinding as possible in general. But there's one thing you overlook, which leaves your argument vulnerable:
It's possible that hiring the best and the brightest is not the wisest move. It's possible that it's a good idea to hire those who are similar to us. I don't think so, but it's possible, we don't know.
To take a concrete example, take the study that showed lab assistants were rated more poorly with a female name on the resume. That's solid proof of gender prejudice. But playing the devil's advocate here, we don't know that it's unjustified prejudice. Perhaps the people evaluating the resumes have had tons of lab assistants of both genders, and found a clear tendency that the women performed worse than their resumes suggested, and the men performed better.
Thing is, even if that were true, I'd advocate for blinding. It's not for efficiency's sake that we should end discrimination, but for justice's sake: You didn't choose to be born a woman, you deserve to be judged on individual merits. Even if women on average were awful at this job, that information should be off-limits to use in hiring decisions, because using it would be a great injustice to those who are not awful.
This is of course even more salient in the case of race and ethnicity. Because while it is highly implausible that women should be worse lab assistants, we do have crime statistics, and if people were allowed to discriminate based on those, it's quite possible that a shop owner could "reasonably" deny Roma entrance to his shop, for instance. It will probably reduce shoplifting! But it's also a horrible injustice to those Roma who do not shoplift. It doesn't matter if that is 90%, or 10%. It doesn't matter if there's just one honest Roma in the whole country. No individual should answer for the statistical proclivities of a category he didn't choose to be in and can't even escape.
But this also shows why blinding yourself to information about race and gender can't just be a "best practice". That asshole shop owner who denies Roma entrance to his shop, he's doing a great injustice, but he might well a comparative advantage over more fair shop owners. Being just can be costly, and because of that, it's important that we demand sharing that burden fairly. We can hope that when we do, we find that it isn't so costly after all, maybe it's even a net benefit. But we must never base our demand for justice on such hopes. Justice first, then profit.