So I finally went ahead and started the AM free trial today. I added a couple dozen tracks to a playlist on my iPhone earlier today, and am just now picking up the iPad to use Music some more.
First off, even after turning iCloud Music Library on, and toggling it a few times, after an hour or so only 14 of the 63 songs I added to the iPhone Apple Music playlist have appeared.
Also, to the OPs credit, I literally JUST ran into the lag bug, and immediately had an AHA! moment...lol. So that pretty much confirms the memory leak, or whatever is happening, is a direct correlation between using Apple Music, and having any pending music/libraries syncing from iCloud Music Library.
Even now, an hour later, still only 15 songs have synced out of the 63 showing in the library on my iPhone. I made one Apple Music playlist just to test it out. Haven't even bothered to open iTunes on the Mac yet, as I have a ton of unmatched songs that will have to be uploaded still, probably creating even more trouble.
Lastly, I decided to snap a quick video of the lag, just as the OP did...ONLY because, if you notice...no music is even playing when this happens...there isn't even a song paused...which is absolutely mind bending! No matter what app you switch into, including the home screen, there's tremendous lag for about 2-3 seconds.
Oh, and has anyone noticed free space being used up on the iPhone? I have a 16GB 6, with about 1.5GB free at all times so I can shoot videos if necessary...but earlier while adding songs to the Apple Music playlist ( or making them offline) I got the 'running out of space' message. Oddly enough, under Usage, Music is at the bottom of the list, as I have no actual music on my iPhone. Been digging around the threads for a topic on this...but anyway, Jayson - you're not crazy after all