No way... There's something up with that. If you haven't synced it to your account through iTunes then that's probably the reason but even then...you should be able to to log into your account through iTunes on your phone and that should be that, all your music should be readily available. Apple Music and iTunes are two separate entities. As in...if you so please (like I do sometimes to support the artists I like even further), you can be subscribed to AM, download an album and have it on your phone, and then 'buy' the album separately on the iTunes store; or visa versa, buy the album beforehand and it'll be fused in with your AM collection when you subscribe and download music at a later point.
I've never encountered anything like that. Try going under Settings>Music and turning off 'Show Apple Music' and/or go to 'My Music' and towards the top there's the tab to show either 'Artists/Album/Songs' etc., tapping on that tab will bring up the options to show the aforementioned, and at the bottom will be an option to 'Show Only Offline Music'. I'm not sure if that'd help but it's worth a try if nothing else works.
It sounds like a syncing issue, though I'm not sure why it's not showing your bought music if you're logged in under your iTunes account on your phone. Try the syncing to your 'hub', and after that try the aforementioned Settings tweaks and if all that doesn't work...I'm not too sure what the issue could be. But they're definitely separate, my dad just got a new iPhone a few weeks ago and I walked him through the steps to download music and whatnot and he was able to download all his old music he'd bought on iTunes years ago right to his phone without being subscribed to AM. Hope any of this helped at all... Good luck!
On the topic...I'm not sure if it'd be applicable at all since you have Android, but I myself love Apple Music a lot. There are zero restrictions on downloading, streaming, and out the 150 GB of music I currently have, about a quarter of that is from finding new music through AM and I've encountered maybe one or two indie albums from bands that aren't really well established yet that aren't on AM.... Oh! And fucking Tool. Of course.
But as far as music available as well as downloading it, AM has been much better for me than any other service. I talked about this a while back in the now defunct AM thread I started but back then I hadn't actually tried any other services and now that I have, they're great but AM is the one for me. Now granted, I've been on iTunes for a very long time and have music through them already, but the fact that I could have that collection I've bought, then have Apple Music just to explore and download music I wouldn't normally try, along with having a 'radio' that is based off of music I've tried on AM (or just going in blind and listening to whatever), download an unlimited amount (er...as much as my device can handle, rather), have playlists that are integrated from already created ones (by other users on AM, professional music peeps, 'recommended', etc.) to ones that I have specifically through AM both online and offline...I really have zero to complain about. Nothing is perfect, but this is pretty damn close to all I could ask for in a 'streaming' service, and it really goes beyond streaming when you get down to the nitty gritty. Pretty much everything I have from AM is now offline simply because I'm on the road a lot these days and don't feel like using even the minute amount it takes through my network, but having the option do to both at my leisure without any restrictions is supremely convenient. Last night right at midnight when it released I downloaded the new Live at Royal Albert Hall concert from Devin Townsend without a hitch, integrated my favorites into an already created offline playlist and it automatically downloaded the songs to offline since I had the entire playlist as an offline only one. It's just really nice. *Wipes off Apple's smile from face; slurping sounds ensue*