It's not a zero-sum game here. It's in Chychrun and the team's interest for him to do well. If he does, his price will be met and he'll get to go to a competitive team. If he doesn't, it backfires for him and it becomes not worthwhile to trade him at all, so he's playing poorly, on a team that doesn't show off his full potential, and he's killing his years before his best chance to secure what is likely going to be his career's defining contract when he hits free agency in a few years.
The team wants him to do well for obvious reasons, and if he is still determined that he needs to go elsewhere, they'll trade him and get some value. It makes no sense to trade him because he's pouting, but not producing. That just shows every other player coming through that you don't have to commit to playing on the team (I'm fully aware he's put up with a lot of poor play already on the team and that's why he's in the situation, but you don't want that to become commonplace now that the team is fully committed to building the right way).
Look at this from the perspective of a general manager as well. You do not want to gain the reputation that other teams can just wait you out without some sort of stimulus such as actual games played, a trade deadline, or injuries. The price is currently set, and nothing is going to change that from the Coyotes' perspective. Long after this is over, Armstrong doesn't need it hanging over head that he isn't firm on his desires for a trade and you can just wait him out.
In the end, no one is looking to spite the other side. If a trade is to be made, it has to make sense for all parties.