The act of surveilling someone is a form of search (Carpenter v. United States (2018)
You are incorrect. What the Carpenter determined, was not about generally "surveilling", but the particular act of requesting historical cell-site location records from wireless carriers.
The Court's majority did rule, that the 4th Amendment protects reasonable expectation of privacy, and the collection of the particular records violated this. But not in general case. What's between your cellphone and the cell-tower can be expected — reasonably to stay private. (Though personally I disagree even with that — anyone carrying an active radio should not expect it to be unnoticed.) But a visible presence in public — detectable by unaided human senses — definitely not. Police have tailed suspects for as long as law enforcement existed — and Bill of Rights' authors certainly knew about the practice existence — and there is nothing wrong about it.
Further, they didn't extend the "third-party doctrine" to the particular case of cell-site records, but they didn't repudiate it completely either.
While the average citizen is allowed to record anything they see in public [...] When the government does it, it is considered a breach of an individuals 4th amendment rights
Nope, it does not — not per Carpenter. Maybe, that's how things should be — though I'm not certain even of that. But it is not, how they are. And the list of earlier court-decisions (which I already posted twice) — the Knotts in particular — shows it.
Further, I was not talking about "the average citizen" — I was talking about a police officer. If one can watch you — without a warrant — he can also record you.