I mentioned it in another post, but I doubt that Bidwill will pay enough cash to convince proven studs to come to Arizona. No, the players don't decide where their current team trades them to, but they do have the power to reject contract offers from the new team, and it happens fairly often in bigtime trades that the player and new team have the extension ready before executing the trade. Three of the four players you mention will want new contracts after next season. Many teams will not give away draft picks for shortterm rentals, and thus, suddenly the player do have some power over where he goes.
My hope is that Ossenfort already now calls the Giants and ask if Thibodeaux is available. Most likely he is not, but certainly they just paid Burns big money, and Thibodeaux has not been great in his two first seasons. Last year he got 11.5 sacks, but only two of them came from beating a starting o-lineman one-on-one. The others were against backups or clean-ups. And by all accounts he took a step back from his rookie season as a pass rusher. After this season, the Giants probably get rid of Daniel Jones somehow, which will leave them with between 22-47 million in dead money, and a lot of wasted cash. Again, Thibodeaux is on a cheap rookie contract, so it would be silly for the Giants to let him go, but no risk it - no biscuit, as BA would say. Who knows, maybe a couple of draft picks (including something like a 2nd), and saving a lot of cash would convince them. From the Cardinals perspective, certainly Bidwill would like to take on a rookie contract for a year or two.