"It's better to say that most songs COLLECTED in the early 20th century were sung unaccompanied, etc. "

But again, MOST of them were half-remembered pop songs from the 1800's, which would probably have been performed with accompaniment.

See for example My Johnny was a shoemaker, written c1859 in the US by a composer and performed in a tour around Europe by his wife, presumably accompanied by the composer on piano.