Zipporah's circumcision of Gershom is (it practically goes without saying) an extraordinary circumstance.
Actual circumcisions recorded are rare in Scripture. But when someone is named under ordinary circumstances, like Abraham, he is a clear mediatorial figure.
"Joshua" is said to have circumcised all Israel, Jos.5:2-3. Obviously, he didn't do the whole job. But he oversaw it. And the only persons who would have had the authorization to assist would be those already circumcised (just one criteria, not a "sufficient condition"). And there is no doubt about it--the Levites, unlike the rest of the nation, were most certainly circumcised in the wilderness. No service at the Tabernacle could have gone on without performance by the ceremonially clean. And we know that the sacrifices were not suspended, the holy things were toted about the waste with the people on the move.
Luke 1:59, "So it was, on the eighth day, that they came to circumcise the child; and they would have called him by the name of his father, Zacharias." Who are they? Covenant representatives; I suspect that the priests were the designated circumcisers, although we can speculate about some other person (perhaps a Levite?) was delegated to this duty.
Bottom line, circumcision was an official act, and official acts are to be performed in the name of the polity. Extraordinary circumstances excepted. "Exceptions do not make the rules."