> it feels like a violation of my privacy

There is no privacy while taking an exam in the classroom, which is part of the point: Exam is taking in full view of examiner (and others) in order to prevent cheating and to make it public to other students (to build trust in the fairness of the exam).

If the exam is taken at home because of extraordinary circumstances it is reasonable to implement measures to replicate this and as it happens I don't think that there is an alternative to software with webcam.