Kessel has used his NTC to force a trade twice now, once when tried to use his trade list to effectively make a full NTC (which backfired and Pittsburgh traded for him) and once when he forced a trade to Arizona. There's really no reason to expect that he'd operate any differently right now. He already nixed a trade to Minnesota last off-season, and that's with him being from Wisconsin and playing for the University of Minnesota in college.
I think he's definitely riding out his contract in Arizona. Arizona is only paying him about $5.1 million in real cash for both this year and next year (and they've already paid him $4.25 million for this year) and I think he has value as a sell-able face for Arizona. I think the soonest he could get traded is as a rental in 2022.