“Can you please explain then how Chassidus or Chabad came about?”
Reb Moshe said he didn’t know their heter to be mevatel their minhagei avos. It’s not so simple that just because they exist, it must be mutar. The original generation of Chassidim who were mevatel minhag Ashkenaz were most likely over an issur (depending on the Chassidus; some are probably not a real shinui).