I do think you may be understating the amount of leverage a team has if they control a player for 4 more years. The team can certainly tell the agent, "Yeah, I know he wants to play in city XYZ, but the have shit for assets and I'm not going to lose my job by dumping off my star player for a bunch of crap. Come back to me with something realistic and maybe we'll talk. Otherwise, wait 2 years".
Still, my uneducated guess is that the situation is unlikely to ever get to that point. A player that signs a 5 year extension with life changing money is probably still quite happy he got that cash, and may not be looking to get out right away. And the team can certainly wine and dine the player to their heart's content; there is a reason Brad Stevens consulted with Tatum and Brown before hiring Udoka. EDIT: Also, the good agents know which teams have the right assets and how those assets match up. So they are unlikely to force the issue if the assets don't exist; they'll advise the player first which teams have a realistic shot of acquiring their client.
Remote possibility is probably the right category for a player like Mitchell. Unless the Jazz really covet an asset Boston has, or covet an asset they can get as part of a multi-team trade with Boston, the trade is not gonna happen even in the unlikely event Mitchell were to demand it. Besides, the Jazz are a good team and certainly will contend in the Western Conference again, no matter what the Lakers fanboys think.