No, you really have this backwards. Price did not sign a contract with an insurance company, nor is he being paid by the insurance company. He signed a contract with an NHL franchise under the NHL CBA, which obligates said NHL franchise to pay out every dime owed on the contract as long as Price abides by the terms of the NHL contract. In his case now that he's on LTIR, that essentially amounts to attending annual appointments with the team doctors to check the box that says the NHL franchise's doctors deem him medically unfit to play.
I am not being pedantic, it is an important distinction. The Canadiens have owed him every dime of the contract under the NHL/NHLPA CBA since the second both sides signed it, and Price is only obligated to uphold his end of the NHL contract which involves standardized medical tests that are all agreed upon under the CBA. Insurance is not mandated on NHL contracts, it's entirely up to the teams to buy insurance based on their own risk tolerance. A lot of teams do not insure contracts, and even the ones that can afford insurance usually only get it on a few big contracts. The NHLPA is not going to allow teams to arbitrarily force a small subset of players to jump through hoops to be paid simply because the NHL team decided voluntarily to insure their own liability to the player.
In the real world insurance companies absolutely have a lot of pull in these situations where a company is paying out an injured worker, but the NHL isn't the real world and the NHLPA isn't a regular union.