purchased music isn't showing up in my iMovie music library for use.

Apple Music songs are DRM protected and cannot be used in iMovie. Downloading the song, even to iTunes, will not remove the DRM protection. Also, Apple Music is a subscription service that provides a license to stream and listen to the music for personal use for as long as one maintains the subscription. That purpose would be thwarted if one could, for example, download the song to an app and then cancel the subscription or let it run out.

Songs purchased from iTunes, however, as opposed to Apple Music, can be used in iMovie, unless in the unusual case that a particular song is DRM protected. If the iTunes song is stored in the cloud, iMovie will not be able to read it, and so one first must donwload the song to the physical iTunes library for it to be available in iMovie.

-- Rich.