Go to phone Settings -> Music -> toggle off Apple Music.
Boom no more annoying pop-ups and it defaults to Your Library in search.
I subscribe to the (under-rated) iTunes Match service [0] which matches all your music in your library with the Apple catalog version and syncs it across all your devices. It will also 'upgrade' poor rips to the official DRM-free AAC 256kbit version. Turn on “Sync Library” setting and it syncs with the desktop Music.app fairly seamlessly.