I think it's less about what Bettman wants for himself and his legacy, and more about what the other owners want since ultimately he works for them. And I just don't get the vibe that it's much of a thought for the owners at this point. They seem more concerned with having a stable group of owners and in growing revenues (largely via a bigger USA television deal, which Quebec doesn't really help with). Plus, the threat of a team moving to Quebec can be leveraged to help them get new arenas that are more publicly financed. Dare I say it might be a more strategic business move to not put a team in Quebec....
I get what you're saying, but I think QC would be welcomed in (once they find another owner/market out West) for the same reason the owners really didn't care that NYI was a financial disaster for 20 years, Arizona and Florida have fallen somewhere between disaster and "functionally poor team." The problems in a market are that owner's problem.
The need for a "they could always move to..." leverage isn't a big deal.
1. There are 14 markets in the US top 35 that don't have NHL teams, so it's not like there aren't places to talk to.
2. Unlike the 90s, teams aren't playing in non-modern, out of date arenas. Arenas and stadiums built in the 1960s to mid 80s were all about "get people in, sell them a hotdog, popcorn and beer, get people out" and anything built after Camden Yards opened is about revenue generation. Most the league opened venues between 1993 and 2003. The fifth-oldest (about to be third-oldest) venue in the NHL is SAP Center in San Jose. It's 26 years old and still a great venue. There's no calls to replace it. We're like 15 years away from these teams needing new venues. Really the only team playing in a venue that needs to be replaced is probably Anaheim.
3. So, let's go down the list by oldest venues: 2 NYI and 3 CAL just got arena deals done; 1 NYR, 4 ANA, 5 SJS, 7 CHI, 8 VAN, 9 BOS, 11 MON, 15 PHI, 16 WAS, 19 TOR, 20 LA... they aren't leaving those markets. 10 TB extended lease to 2037, 12 NASH can't leave until 2049.
The only teams that can play that card of relocation anytime soon:
6 STL, the city just lost the NFL, Blues will get an arena deal done. I can't find their lease details.
13 OTT
14 BUF (owner also owns the Bills, so it will be a package deal)
17 FLA - 2028 lease expires