Guys like Fisher and Neil are considered "greats" in the organization because of the countless hours they put into the community, and how over-and-above they were with the fanbase. If either of those guys just showed up to the rink to play and went home after the game to live a private life and didn't involve themselves with the fans or anything else, would they still be considered "greats"? I highly doubt it.
I mean, you just named probably the two guys in Sens history who put more into the community and fanbase than anyone in the history of the franchise. If those two guys had been selfish pricks, there's no way they'd have the legacy that they do.
Fisher and Neil aside though... I think all NHL fanbases have a thing for "blue collar" type players, because working-class fans can relate to that kind of work ethic - a guy who is in the league because he works hard, not because he was born with all of god's gifts. "He might not be the most skilled guy, but he overcame that and made something of himself anyways". I mean, that's the most relate-able story in sports, right? It suggests that anyone can do it as long as they work hard enough, which mirrors "the American Dream" so to speak.
Parents can push guys like that as role models to their kids, because those players reflect the kinds of values you want your kid to grow up appreciating, regardless of their actual hockey skill - you can point to a guy like Mike Fisher and tell your kid that he embodies the kind of attitude that helps you in life, not necessarily limited to hockey. Then, those kids get older and, having grown up being told that those characteristics were virtuous and looking up to guys like that, appreciates the same types of character as adults. They then pass that down to their own kids, rinse and repeat.
Can you name a blue-collar hockey player in the league, on any team, who donates huge chunks of their free time to the community, and is a genuinely likeable & charitable person, that isn't loved by the fanbase?