No there wasn't an illegitimate stage, but legitimate stages, were considered as Broadway, where Drama and comedy and musicals would play.
The other stages back then, in the turn or the 20th Century, were Burlesque and Vaudeville. The former being mostly girly and comedy shows, and the former, being Comedy, acrobats, animal acts etc. Many of the biggest names back then, were in Vaudeville, like Bing Crosby, Bob Hope, Red Skelton, Jack Benny, and most of the comics from that era.
They later on, moved from Vaudeville, to Radio in the twenties, and then on to Television in the forties and beyond.
So you could consider those two types of stages, Illegitimate, but that makes them sound, like they didn't belong. Were as they did.
They were the main entertainment for millions of Americans in those bygone days. All before movies, radio and other mass entertainment.