It's been proposed in the past:
en.wikipedia.org
en.wikipedia.org
Obviously, it's never happened.
The original WHA was able to succeed (for a time) because of a couple of factors. First the NHL was quite small. By 1972 the league was at 14 teams (and then 15, the Islands, specifically to block the WHA from New York). But there were still a lot of major cities without teams.
As well due to the reserve clause the NHL paid its players very little at the time.
So the WHA could succeed by putting teams in major cities and were able to lure out major stars like Bobby Hull by offering more money. But even then, the league only lasted a handful of years.
So it's hard to see what advantage any new league would have over the NHL. I mean maybe is Saudi Arabia came in and spent just unreal gobs of money to lure players they could do something (I'm thinking of LIV golf here), but otherwise you're going to have a bunch of second rate players playing in second rate arenas.
The only point of differentiation you mentioned is doing it the "right way". You mentioned 3 points for a win. Casual fans don't give a crap about that. Beyond that, what is there? What would cause someone to watch the new league over the NHL? I dunno - more fighting (but concussions)?
I'm just not seeing any point of differentiation. There have been NFL competitors in the past - USFL, XFL, and UFL - but at least their point of differentiation has been playing at a different time of year and not direct competition (except briefly with USFL 1.0, which quickly led to their demise).
So maybe you go "well what about just an independent but more affordable league"? Again that's been tried before as well.
en.wikipedia.org
You can see that the IHL was an alternate farm team to the NHL, but in the 90s flirted with the idea of trying to compete directly - it didn't work.