Two points, one nitpicking.
1. Recuse refers to when one bows out of judging something because they feel uncomfortable doing so for some reason (typically an appearance of impropriety). One is not recused by someone else as the word is used in the subject line. If a person is, by rule or policy or simply one's boss, prohibited from judging something (a critic being a judge in this context), then they are removed, unassigned, replaced, etc., but not recused.
2. I do not believe that the examples mentioned in this thread would constitute per se reasons for removal as a conflict of interest. My guess is that the chief critic makes this call, perhaps with input of the editor, but that it is not a removal (or a recusal). Were every critic who has worked with, or is friends with, etc., someone significant in a production, prevented from reviewing their work, we would not have a lot of reviews that are well-known to many of us, going back (within my knowledge) to Atkinson who would have written just a patchwork quilt.There are cases of actual prohibition but I would be very surprised if that's what is happening here.