I don’t think that the general public (i.e., those who have a superficial familiarity, if any, with classical music) is susceptible to embracing any more than they’re already aware of, but musicians can certainly be encouraged to explore. And thanks especially to the internet, one doesn't have to be in an academic setting or have a conventionally “music-educated” background to do that. For example, a number of YouTube channels are dedicated to music of lesser known and less acclaimed composers.
I know it can seem as though the big names are obsessed over, but I think it’s because their output is, after all, of such high quality; their reputations (and, yes, “fame”) are well-deserved. But a significant number of aficionados really do like to get off the beaten track, too; maybe we think we’ll find an underrated masterpiece, perhaps we’re just curious about what the also-ran contemporaries of The Great Ones were doing, and sometimes it’s just nice to hear something fresh.
Promoting interest in the relatively obscure can result in renewed awareness of historically significant composers, and sometimes old masters are rescued from oblivion—among the small subgroup of people who care about such things, anyway. It’s a niche, but then so is classical music generally—like so many other areas of the arts in an increasingly coarse culture.