>That said, why hasn't Google yet created a desktop application for Google Play Music?
I don't understand the need (and I guess that Google does not either).
Is it because you need an offline mode ? That's the only addition I can think of.
With the play music chrome extension, the website responds to media keys, so outside of a dedicated app icon in the tray (even though the miniplayer has one), I don't see what a desktop app would bring.