This is the best and least frustrating way to roll.
Way back when, I tried some other suggested fixes, like starting anew and enabling iTunes Match first. That resulted in a near duplication of my library. Turns out it isn't so easy to start anew: synced libraries are retained with your Apple ID for a specified duration. I wasn't about to create a new Apple ID. What a pain.
I worked with Apple engineers and demonstrated (and they replicated) deletion of files on my file system. I know that some on this thread don't believe that's possible but it certainly was and did happen.
Regarding songs working in iMovie: I had a vast percentage of my songs labeled as "Apple Music" not as "Matched". This songs in my personal library became indistinguishable from the AM library. It was these songs that could not be used in AM.
I finally dumped AM and iTunes Match and waited for the arrival of iOs10. I have re-enabled AM but. It match. I still have tons of duplicates in my library and other oddities, like where AM does the annoying thing of matching a fraction of songs on an album, then filing them under different metadata.
Fortunately, I did discover that you can edit metadata of AM supplied tracks, at least allowing one to reunite tracks as a single album.
I do find Music much improved on iOS10