One thing I've found myself doing to my detriment, is try to figure out what percentage of people feel a certain way on an issue without basing it on polls or studies. It's human nature to look at an issue, consider one's own perspective on that issue, and presume that more people agree with you than actually do. I tend to lean left on these things, and if I were to guess how many people would be somewhere from mildly relieved to actively happy to make a change and get this issue off the table, I'd almost certainly overshoot that number. I suspect someone on the other side of the issue would likely do the same. (Not saying you are or aren't, but I feel it's helpful to remind myself that this can become an issue.)
I would like to see some polling on certain specific options and outcomes (Chicago replacing their logo with the cool hawk's-head logo concept, the Atlanta Hammers, the KC Chiefs adopting a fire station motif), to see if specificity helps move people in one direction or another. That would, potentially, be illuminating.
My preference would be to rip the bandaid off, make the change, and get everyone mad all at once so we can get over it and back to enjoying sports. I'm sure there are others who feel there is no bandaid at all, and would rather I just remain annoyed/disappointed about the whole thing forever. Ultimately, it will come down to the almighty dollar; when and if the owners of the teams in question decide it's in their financial interest to make a change (or, more accurately, when they feel remaining with the current name is too harmful to overcome), they will. Until then, they won't.