And yes, this is actually a well-known problem in Italy - with (typically male) professors being routinely accused (and occasionally convicted) of favouring attractive (and typically female) students.
If I say something imprecisely or if I make a non-fundamental mistake, an oral setting gives me the chance to correct myself and prove to the examinator that I have a strong grasp of the material regardless.
Written exams, especially multiple choice and closed-answer quizzes reward people who regurgitate the notes, oral exams and written long-form open questions reward actual knowledge.
Of course the "better" methods require a greater time investment, and I can't really blame professors who choose not to employ them. But it's quite clearly a tradeoff.
This is just even further proving the point, which is that in an oral context this means that the animosity of the examiner is much more significant than in a written one, which by definition implies that the oral one cannot be fairer than the written one.
You yourself are saying that you "have the chance to correct yourself". This is either because you will self-correct yourself on recognizing a specific (perhaps subconscious) face or gesture from the examiner, or because the examiner will directly tell you that you are wrong. Both cases present ample opportunity for unfair discrimination. In the first case, perhaps a person is less skilled at reading people, or perhaps the examiner just has a better poker face. In the second case, you are now at the whim of the examiner to decide based on your body language whether "you are making a non-fundamental mistake and deserve a second chance" or just "have no idea of the material and don't deserve a second chance". And, compared to the written exam, there is absolutely no record of the context that drew the examiner to such conclusion -- which is also kind of important, since evidently the written exam is also subject to some discrimination.
Nobody expects you to be 100% on point, it's just impossible; it's not like the spoken variant of a written exam. The kind of "correction" I mean is more along the lines of what would happen during a normal conversation. Imagine I was asked to write a recursive algorithm and I forgot the base case. It's not a fundamental mistake, but the professor might interject to make sure I actually know about termination, inductive sets, etc., which is actually great if you understand the material deeply, because it gives you a chance to prove that you actually just forgot.
Obviously this is assuming good faith by the examiner, but if you aren't willing to assume that, there aren't very many examination formats that are going to work very well.
How you deal with that is really where your score comes from. Because if you know what you're talking about you'll correct it and while doing so show that you know a lot of related things. While if you have no idea you can't guess yourself out of that type of question.
How would you blind oral examination so that the examiner is unable to distinguish the student’s gender/race/identity?
FWIW, the study that "proved" that appears to have been a pretty bad study. So, in reality, no: people are not terribly prejudiced, and things don't change significantly when you blind the examination.
I still think we are conflating the objectives of
1. Teaching students skills and knowledge they need and want, and
2. Rationing access to jobs, status, etc.