Briefly, the bill of rights (Amendments 1-10 of the U.S. Constitution) puts limits on the power of the federal government, and the 14th amendment extends these limits to state governments.
These amendments say nothing about agreements between private parties.
By way of example, if a production company hires someone to perform in Hamlet, but the person then gets on stage and launches into a soliloquy about how much they love the organic dental floss they just bought from their neighbor Moon Unit, the production company can fire them because they failed to do the job they were hired to do.
Is Rogan the congress of the United States? No.
Is Spotify the congress of the United States? No.
Is Neil Young the congress of the United States? No.
Is Rogan a state government? No.
Is Spotify a state government? No.
Is Neil Young a state government? No.
We should all take a moment to be grateful that the answer to all of the above questions is "no."
The 1st Amendment could (at least in theory) come into play if one of the parties asserted a state or federal law against the other party and if the law in question did in some way limit speech. So far, I have not heard that this is what happened in the Rogan/Spotify/Young saga.
In certain instances, courts will refuse to enforce agreements between private parties (e.g. between drug dealers) on tehe basis that the agreement itself is against public policy, etc., but this is not a constitutional issue.
Roughly speaking, 99.99999% of disputes between private parties do not involve constitutional issues (okay, I'll admit I pulled that number out of my tailpipe, but you get the point).
