There are other poems of Yeats that were specifically written with tunes in mind, and not just in the early poems.

A Full Moon in March", published 1935, is full of them: "Three songs to the same tune", "Two songs rewritten for the tunes sake"; "Supernatural songs"; and of course Last Poems, published after Yeats died, is full of songs.

As for "The Salley Gardens", of course it uses an existing song. It distills it into something stronger and simpler, and more spare and subtle. It's the same thing that the folk process can achieve, sifting out whatever is not needed.

(It's the same thing that Francis McPeake did with "The Wild Mountain Thyme", and that is why people miss the point when they point to the earlier Scottish version and imply that this somehow reduces the value of the distilled version.)