In a written exam they can cover the names - give you a random number as you enter the room and you write that on the paper, and but your name and number on a different paper. You also need to type everything out on a computer with spell check. (and even then if you write bucket or pail will identify you but it is unlikely any professor knows you well enough to tell those)

When you audition for a symphony you perform behind a curtain and are required to wear soft slippers (so they can't tell if you are a wearing high heals - female).

We can probably use voice changers so the examiner cannot tell who you are by your voice, but those tend to be fatiguing.

If you can't trust a professor to professionally and impartially grade someone's work, the system would likely collapse. This is not to say that there hasn't been cases where professors have been shown to be biased, there has. But the premise of universities is to give professors some autonomy in the way they teach and evaluate students.

No system is 100% the question is are we good enough. As a white male I haven't seen many problems - but also because I'm in the group least likely to see one.

True but I think there's still an element of falsifiability to a teacher's evaluation of an essay that doesn't exist in an oral exam or interactive discussion. An essay is an artifact and if a teacher is giving student A worse grades than student B, a third party can look at that artifact to see whether it's remotely reasonable. A 1:1 discussion or an oral defense is much more subjective.

Not saying this is a fatal flaw, but there is a bit of a tradeoff there.