I came here as I had the same question and may have found a fix after reading this on the link given by ed2345:
"If you can’t see iTunes Store, choose Music > Preferences, click General, make sure iTunes Store is selected, then click OK."
I wondered what would happen if I unticked iTunes in Music>Preferences, and realised that I could also untick Apple Music, so I did both, quit the app, relaunched, and when I clicked the Search box it showed 'Searching Your Library' in the middle of the screen.
I was then curious to see what would happen if went back and re-ticked both boxes, so I did that, and now, after quitting the app and reopening it, when I click in the Search box it shows all three options in the top right.
After various searches using all three, and moving between, Artist, Songs, Albums etc on the left, to test its resilience, it appears that these three search options are now permanently fixed and selectable.