They don't have any authority over the security check itself, just the line, as delegated to them by the airport. As you say. But the airport is where you line up, the airport is who you pay for your experience (XF, $4.50 per ticket at SFO anyways), and the airport is private property.

So I guess my question is, why shouldn't the airport - who you're paying admission to - get to decide who can line up where? Why should they not defer that authority to someone else? The airport is a business too, generally speaking, and they have customers who value getting through the line faster. They've deferred that determination to a third party who also manages said line-up.

What difference does it make who's running the line - SFO staff or Clear staff? They're going to be private contractors one way or the other. And at SFO you're just going from one private contractor who manages the line-up (Clear) to another one who manages the actual security (CAS, not the TSA) - to another private company who flies you somewhere. Unless you stop at the privately operated restaurants or lounges.

SFO is owned by the city and county of San Francisco, but that's beside the point.

The TSA should run its own line because it's the one imposing the screening requirement.

The TSA doesn’t operate the checkpoints at SFO, it’s a private contractor called Covenant Air Security.

You are right then that SFO is public property but of course that doesn’t make it freely accessible.