Diversity aside, I may or not be an outlier here because I like the same characters being exposed to new situations over time. I want Peter Parker to be Spider-Man, and it bugged me when - for prolonged periods of time - there was no comic book published by Marvel where he was. I was disgruntled when the 90s reboots took Hal Jordan out of the Green Lantern book; it was irrelevant to me what race or gender his successor was though.
But then again it used to be we got a reboot maybe once a decade; usually longer than that. A reader could go over the story arc of Tim Drake becoming Robin in the eighties and stay twenty years with the character until there was a very good sense of continuity involved. In a sense it allowed a reader to bond with the characters. These days... well, it seems to be happening constantly, it's hard to keep up. Is the young Cyclops brought back from the past (along with the rest of the original X-Men) still the one in the current universe? Is it someone else? I don't know. I don't even care who has what name; it's irrelevant to me if Thor is a girl, as long as Thor - the character - is around and involved with the Avengers.
The other thing that bugs me is how much comics are used as billboards for movies. I get it, that's where the big bucks are... but knowing there's a beef between Marvel and 20th Century Fox or whoever because the latter have the Fantastic Four rights, and that's the reason there's no FF comic being published which in turn affects the, well, IC part of the universe's continuity and takes loved characters like Reed and Sue Richards out for the duration annoys me. It's not a creative reason, they're not trying to tell a better story this way... it's just bullshit marketing politics.