Some of this is now almost ancient history, but the league did more than that to keep Arizona.
Remember the league was paying the team's bills back in 2008-2009. The owner planned to sell the team to Balsillie, but the league blocked him in court and went so far as to purchase the team out of bankruptcy. That was a lot more then they did for any other team.
Balsallie would be the owner of an NHL team if he played by the rules. Balsallie was his own worst enemy. IIRC, he first tried to buy the Penguins. Then he tried to buy the Nashville Predators, promising they would stay in Nashville. Balsallie broke that promise, indicating he wanted to place the team in Southern Ontario, and the sale was voided.
Then he bought the Coyotes. Problem is, he went out of his way to antagonise the NHL owners and Bettman, by giving them the middle finger, and made it clear that he was going to move the team to Hamilton, and even went as far, as promoting a season ticket drive to Hamilton area residents, even if the owners had not even ratified the sale. At the end, the owners refused to validate the sale, and took over the team themselves, at which point Balsallie sued the NHL.
This happened before. In 1983, Edmonton businessman Bill Hunter had bought the St.Louis Blues from Purina Dog Food (who owned the team), and had announced that they would play in Saskatoon. The NHL voted 15-3 against the sale, since Saskatoon had no NHL suitable arena at the time, and they viewed the Saskatoon market as too small for an NHL team. Incensed, Hunter and the Blues refused to take part in the 1983 NHL draft, and walked away from the team. The NHL took over the Blues, until an owner eventually was found.