Whole point of the digital face is to look real though, and freezing the gaze would look unnervingly fake.
For example, you could then use the AVP to stare at people and then claim afterwards you were doing no such thing.
I'd suggest blurring the face in a "password input context" (like password fields on the web with their redacted display text), but I suspect that that'd go against what Apple wants the Vision Pro experience to look like.