The reason that your Apple Music does not appear is that Apple Music songs cannot be used in iMovie because they are DRM protected. Downloading them to iTunes will not remove the protection. You can still purchase and use non-DRM protected that you purchase, or have purchased, directly from iTunes.
Also, Apple Music is a subscription service that entitles you to listen to the song as long as you keep paying for the subscription. In other words, you are not purchasing the song, but only renting it so to speak. It is not intended that the songs be downloaded used in any other app. Otherwise one could buy the subscription, download numerous songs, and then cancel the subscription, and use the songs without paying the subscription fee.
-- Rich