The most obvious place where people have "disowned" this word is, ironically, on /mlp/ on 4chan. Now whenever they refer to bronies, it's always going to be those weird, crazy neckbeards going to meetups with far too many plushies. For much of the fanbase, they use that word to refer to each other and themselves and it is not offensive at all.
Yet when used outside of the fanbase, it is almost always in a mocking manner. The fanbase is often being associated with awkwardness and whatnot, which is why people who are not as hardcore of fans want to distance themselves from "bronyism".
Watching this show does not make me "hip" or "cool" or "special" and stuff, it's just a hobby and labels are cringey. Notice how Regular Show, Adventure Time, and Gravity Falls have no labels for their fanbases. It really isn't needed at all.
Interesting, but I'd like to point out that it's also hard to give non-weird sounding labels for fans of the other shows.
One of the reasons why the fanbase for MLP:FIM is the odd one out is because aspects of the show allows for easy labeling. The cast mainly consists of a singular, non-human species whose common name happens to be comfortably pronunciable in English - "pony". Furthermore, the descriptive name is solely associated with this particular work and not associated with other similar works. There is no other work I know of that revolves around ponies and has a sizeable fanbase.
Labeling a fanbase of other shows, movies, books, games, musical bands, etc. would just be too awkward.
For example:
Adventure Time: The diversity of the characters on this show makes it difficult to give its fanbase a collective label. The only thing they all have in common is that they call the land of Ooo home. Maybe "Oootakus"? Although the Japanese term seems dissonant from the show's American origins.
Regular Show: Similar reasons as "Adventure Time". The commonality all the characters have is that they work at the park. Any label with "park" is bound to lead to confusion (Are you a fan of "South Park", "Parks & Recreation", "Trailer Park Boys", etc.?). Maybe Hamboners, but who the hell wants to call themselves a -boner?
Star Trek: Now this one works. Despite the show's diverse cast, "Trek" is such a short, uncommonly used English word that modifying it into the label "Trekkie" doesn't seem too awkward. Furthermore, it cannot be confused with other fanbases because there's no other work associated with the word "Trek" that has a sizeable fanbase.
The Walking Dead: The cast members are all Georgians and all fight zombies. That doesn't really help with labeling. The title doesn't lend any help either, unlike Star Trek. Calling the fanbase "The Walking Dead" self-refers the work which is a no-no, and calling them "Deadheads" might confuse them with other fanbases (The Ungrateful Dead)
Game of Thrones: Too diverse of a cast. Names from this show are generally too long and awkward to form a catchy label anyway. Title doesn't help.
Futurama: The show takes place in space and in the future, so "Futurenauts"? Meh, not really. "Naut" as in "astronaut" is associated with exploration into the unknown. The cast is just interstellar delivery - no real exploring there (though they do end up in some weird places).