Your hypothetical isn't too far from the truth. As I understand it, under the rules of evidence, the government can use evidence that was illegally obtained, provided it was obtained by a third party. Just speculating here, but presumably the government cannot order such evidence.

To modify your example: Imagine instead that a "real" burglar breaks into a private home and makes off with several valuables. One of them, hypothetically, is a laptop with years of highly incriminating personal data on it. By some miracle, the government arrests the criminal and recovers the loot, including the laptop. Perhaps they're legally allowed to search its data. And viola! What a trove of incriminating evidence against not the burglar, but the home owner.

I'm not sure the analogy to 1st amendment is easily made, but clearly the private-government distinction can be abused, as Gato cites in the article, surely breaking the spirit, if not the letter, of the law. But there are limits to free speech. In the 1970s the KKK (or was it Neo-Nazis) marched through a Jewish area (Skokie IL, if I recall correctly). It was controversial, but it was protected "speech." But I doubt anyone would claim an anti-Semite to have a right to spew his venom from the lectern in a synagogue.