Based on prior case law ...
Without a citation the reference is useless.
... my understanding is that the government cannot utilize the private sector as a conduit to act on its behalf to take actions the Constitution prohibits the government from doing on its own.
In the abstract, that is generally true. However, as applied that can be a difficult to decern. What you seem to overlook is that the government has the same right of free speech as you, me or anyone else. It can ask, it can demand, it can scream, holler and pound the table. What it can't do is coerce, do this or it will do that. Kinda like what happened with Eric Adams.
Here in Connecticut, I was once a Justice of the Peace. But, no JP Courts here (that I know of).
Hey dumbass, what do you think JP means? It means Justice of the Peace, the lowest level court in the US. Dumbass. And congratulations on your service. It requires absolutely no legal training, at least in Texas. A requirement you meet.