This sounds like cutting your nose off to spite your face. The Winter Olympics probably gives ice hockey its biggest global audience so stopping NHL players participating prevents the NHL from raising its own profile. Does ice hockey in Europe operate in the same fashion as NHL? As in with franchises where players sign contracts with the league.
To my knowledge, European club hockey is organized more like football. The players are employed by the clubs, subject to rules set by each country's league and with some international (i.e. IIHF) regulations on top. Whether there is any coordination between IIHF and NHL on that level, I do not know (but I doubt it, since they are organized so differently).
I think the deal with NHL getting their way is that it was established as a big entity long before international ice hockey was a thing. For football, it was the other way round. The World Cup was first held in 1934, long before professional football had the financial muscles to have a say. It's all about who can say "fuck off, no deal" and hurt the other more. To elaborate.
Hockey: The NHL can say to the players, "You go play for your country in the IIHF World Championship or the Olympics if you like, but we terminate your contract."
Football: The clubs can't say to the players "OK, go play at the World Cup/continental tournament (e.g. Copa America) if you like, but then we terminate your contract". Why can't they? Because FIFA would say "You do that and your country are not going to be allowed into the World Cup".
It's about the relative standing of the league vs. international tournaments.
That being said, I think it is sad that the IOC and the NHL haven't been able to find a common understanding, because I want to see a tournament where each country fields the best team possible.