The notion that "integrated" starts with anything in the twentieth century is absurd. There were shows with wholly story-attuned scores in the 1990s--Reginald De Koven's Robin Hood and John Philip Sousa's El Capitan, for instance. And that's speaking only of the American theatre. In England, the musical--meaning a dramatic work with dialogue and song that is more or less in the popular rather than the classical idiom--dates back to The Beggar's Opera, in 1728.
The Black Crook, so often called "the first musical," had only two vocal numbers when it opened, in 1866. Other songs were added in during its many productions, as a poster above has noted. And most of the titles that scholars bring up--Evangeline, for instance--from the 1870s and 1880s were either plays with a few songs and dances or, at least, started that way. Evangeline didn't have a full evening's worth of vocal music till four years after its premiere.
It's also misleading to go by the nomenclature of bygone days--"comic opera," for instance. Gilbert and Sullivan didn't "insist" on that term. There was nothing to insist about; comic opera was what most English musicals above the level of pantomimes were called. "Musical comedy" as a term didn't come into use till after Gilbert and Sullivan had parted company, and the term "musical" (that is, without the "comedy"
wasn't used until the 1940s, to identify a work that was more adult than the silly shows but wasn't an operetta or "musical play," either.