Has Broadway been changed by the presence of shows like Caroline or Change, or Grey Gardens? Would the latter have become popular regionally if it wasn't on Broadway? That's really hard to say.
If Fun Home were to move to Broadway, particularly in a very crowded spring, and compete in the commercial horserace that is the Tony, where commercial viability is the prized asset, what will happen? It may very well pick up an award- for Cerveris, or score or book (or both). Or it won't. There really isn't a great song that can be showcased out of context for the performance.
It's not "family friendly" or a spectacle of any kind. Tourists will have no idea what it is. Older audience members will probably balk at spending $100+ on a show with "no set."
Next to Normal was able to cultivate a strong youth audience, which I can't see Fun Home amassing despite its quality. And, really, at the end of the day, Next to Normal is a lot more conventional and showier. It's a domestic drama with a twist.
With Fun Home's strong reviews, female-centric cast, small size, and relatively few design demands, it will show up regionally regardless.
"...everyone finally shut up, and the audience could enjoy the beginning of the Anatevka Pogram in peace."