There already is unconscious bias. You can see the student's name, their penmanship, their writing style, you likely know who they are, etc. An oral exam just changes things by changing the bias to accent, inflection, annunciation, skin tone, dress, etc.
When I was in school, 95% of grading was blind and nothing outside of exams was handwritten.
And while it was possible to de-anonymise, the academics were all in support of blind grading so why would they?
The only exceptions were projects where everyone was assigned a different topic, and graded presentations.
Of course, this is simpler in STEM subjects - it's not like you can guess someone's race or gender from their switching power supply design. Subjects that prize in-class participation and lengthy essays would probably be harder to blind-grade effectively.
This also depends quite a bit on class size. If you are one of 9 profs and 36 TA's on a whole year of a 1st year intro course, you can get together and batch mark finals very effectively blind.
If you are teaching a 4th yr/masters mixed class of 11 by yourself, you pretty much get to know who is who whether you want to or not. I suppose avoiding handwriting can help if it's appropriate (e.g., won't work on a math course) but I suspect you'll know everyones style by then anyway.
The examiner doesn't even have to use the oral exam to give a grade. They could use that part simply to figure out whether the student passes or fails. It's very difficult to cheat in an oral exam. Combine the oral exam with the written exam and you could get an overview of what the student knows.
One unfortunate thing is that poorly done, orals can be very uneven.