As I mentioned below, I had to fix a script to workaround a problem playing a playlist when only one track had as yet been copied to it, which prevented the rest of the subsequently added tracks from being recognized by MiniPlayer and Up Next; thus, only the first track would play. Similarly, if you were to play (the AppleScript command) any track in a playlist, the following tracks will not play because they haven't been recognized by MiniPlayer and Up Next ("No upcoming songs").
Doesn't go good when Mini Player is empty:
You can play the playlist and everything's OK. You just can't initiate play of the entire playlist by playing one of its tracks with AppleScript.
Goes good and loads in MiniPlayer:
Additionally, I found that if the MiniPlayer was already loaded up with tracks, playing a track (via AppleScript) from anywhere would play that track and then resume with whatever is next in MiniPlayer.