I used to be indifferent but over the past few years, I have gotten a bit of a soft spot for the Leafs. I've made friends with a handful of (nice) Leafs fans, so it's hard for me to not get absorbed into all of that. Plus, our games against them are usually a lot of fun. In addition to that, I think it would be kind of funny to see the rest of Canada shit thsemelves if the Leafs did win (I think there is often a collective dislike for Toronto sports fans). I don't think the comparison should be made with the Rangers, though. If anything, the biggest competition is more of breaking the drought of a Canadian team winning the Cup.
Also, call me crazy but I kind of like the 54 year drought as a part of our history. I realize it's not something you particularly want to gloat about, but it became such a huge thing leading up to the few Playoff runs/hopes, and eventually the '94 win. You think about the signs, the announcements, fans/player reactions, just everything...all because of that damn 'curse'. It was shitty, but it became a historical moment for the franchise. Any ways, I bring this up because a part of me feels pretty protective of this part of our history, so the idea that a random team might break this record...feels a bit invasive? I know it doesn't make much sense, but I guess I'm talking more from a sentimental point of view. I also think that's what happens when you start putting the Rangers' history in the equation, whereas if you didn't, I probably wouldn't care too much.
That said, it does vary. There are some teams I would be more than happy to see them never win a Cup :^).