Thanks for your detailed reply.
You say we need to find a winner, but that goes back to my point about manipulating the scores. We need to find a winner because the current system requires it, but a 5 min overtime followed by a draw takes up the same amount of time and ends with a totally natural scoreline instead of a possibly undeserving team coming away with a full 3 points. It's just as logical, maybe even more so than the compromise you talk about.
Also you mention that it doesn't affect the integrity of the game during the season, but there are 3 points being handed out every time there is a shootout even though a team wasn't able to win during regulation/OT which realistically skews the standings no matter how insignificant. You say it isn't good for the playoffs, but that is undermining the importance of the regular season as well. Again, it doesn't need to be compromised like this.
But the majority of fans don't like draws, you're right, and that is of course the NHL's motive to have the shootouts. But there are other ways that would satisfy the sports purists more I think.
Here's the thing - the NHL doesn't need to cater to hardcore hockey fans. Hardcore hockey fans are going to tune in no matter what - shootout, tie, coin flip, mud wrestling to decide, zamboni race, whatever. Hardcore hockey fans, sports purists - don't need to be satisfied. They're the core - they're not going anywhere. What you and I want is irrelevant, because we're going to watch regardless.
What the guy who goes to four games a year, can't name the third line center and just yells "shoooooooooooooooooooooot" whenever the team has the puck in the offensive zone is who the NHL caters to, and it's who the NHL needs to cater to.
Edit: Also, in that vein - the three point system would probably fail in NA, specifically some markets in the USA, for that very reason. US sports are about W-L records - ties are incredibly rare in football and baseball, and non-existant in basketball. If you bring back an OT loss as one point as well as a tie, you have the ol' 48-10-3-6 line to determine standings, which just causes some people to shut off. The casual fan is just more important.