So it seems like this has become a question of what does it mean to be a franchise?
And what it seems to me is many people are taking the side of "the arbitrary definition put forth by the NHL".
Is a franchise the players? Well if so, most franchises are changing every year.
Is a franchise the front office? Well if so, it's also changing pretty quick in many cases.
Is a franchise the owners? Often this lasts a long time, but it does change as well.
Which leaves location/fanbase. Of all the things about a team that last longest or is the most consistent, it's this.
Take the case of the Cleveland Browns. If the NFL defined things the same way as the NHL, Jim Brown would be the greatest Raven. But nobody says that because the NFL's arbitrary decision was different.
Which fanbase in the NHL feels more connected to the history of the Jets 1.0? To me, that holds a lot more weight than "the Coyotes are the Jets 1.0 because the NHL says so". Coyotes fans don't care about Jets 1.0. As a Flames fan, I wouldn't consider Atlanta Flames stats relevant. Jets fans don't care about the Thrashers. Putting those histories together is meaningless to anyone.
Imo, fanbase and shared history is what defines a franchise