> government cannot hire a 3rd party to do what would be unconstitutional if done directly

Is there actually a legal precedent for this?

There's a lot across agency law, but one example would be Skinner v. Railway Labor Executives' Assn., 489 US 602 (1989) involving a drug test performed by ostensibly private railroads, but pursuant to federal regulations. That was considered sufficient government action to implicate the 4th Amendment.