But private companies can be sued by employees and those private companies could then sue the Fed Gov for coercion?
No. It doesn’t work like that. The President can use the bully pulpit to urge all that he wants.
OSHA was the coercion (the proverbial gun) so to speak. Now it’s just words from the President.
Companies can say “no.” They can say “yes.” They can do what they want subject to law and contract.
The point is that yesterday's SCOTUS decision on the OSHA mandate had nothing to do with whether employers can implement mandates on their own, or whether the President can urge that businesses should do that on their own. It just was not part of that case.