Hello zeevil,
Welcome to Apple Support Communities. We understand you have some issues downloading songs to your iPhone with Apple Music. The steps you listed were great to start with.
We'd also suggest confirming if the issue persists depending upon the network you are connected to. We'd recommend testing downloads while connected to another Wi-Fi network, plus using just cellular data.
Reply back with how your test goes. Also note what available storage you have available. Go to Settings > General > About to find out. This will help further narrow down what's going on.
Take care.