He has one more left on his contract. Then he's a FA. NTC has nothing to do with it.
It does to the extent he can steer to his landing spot. He can decline every trade suggestion that isn't the one or two places he wants to be and if they are lucky they get a Tkachuk return and if they aren't he gets a Hanifin return. But I could actually see him declining anything and forcing them to either re-sign him or let him walk for nothing, knowing they would not let him go for nothing.
If KD had the stones to at least fight for a limited NMC on Mitch's first deal this could have been avoided but he gave away all his leverage. And not to belabor the Willie point but the one way out was for Shanny to allow him to be dealt over the summer. Kyle may have had that safety net in mind with his first deal.
Until July 1 they could still trade Willie if one of the teams on his non veto list thought he was worth that Trelific deal, correct? I am going to guess Willie's "won't go there" 10 was mostly populated by playoff teams that had the cap space and would have had strong interest rather than doormat clubs that would not have dreamed of adding a guy that was certainly going to be a rental for them. It basically turned the MNT into a NM. So AZ/SJ/Anaheim could perhaps be viable destinations? Utah/AZ has their single marquee player locked up so the overpaid Styles wouldn't screw up their salary structure and $15M in LTIR cap is coming off their cap next season. What a splash for the club to make their first year in the new building. They would need crazy deep pockets though because of all those Leaf-only bonuses so it might still be out of reach (go Tre).
Getting back to reality, if the playoffs don't go well they should probably talk to Mitch and explain they have salaried themselves in to corner and his next deal will make them even less competitive. So they are "exploring all options" and see if he has any interest in being god somewhere else rather than second fiddle to Matty forever. Then after he says no they hunker down to wait out the last year of JT's deal. No clear way out except to run it back. They can and they will. Again!