iTunes shows error when trying to play music stored on iCloud Music Library after I delete local copies

In a space-saving exercise I've recently deleted all of the local copies of iTunes music files that I've kept on my hard drive on the basis that they're all on my iCloud Music Library anyway, and I don't need them saved as hard copies (they're all backed up on my external hard drive anyway if I ever lose them).

The trouble with this is that now my iTunes can't play any of the tracks I imported and uploaded. It will show the error "The song could not be used because the original file could not be found. Would you like to locate it?". Clicking on Locate is a bit of a fruitless exercise, because none of my music files are on my hard drive anymore, and clicking Cancel plays the file anyway because, well, the file hasn't actually been deleted. It's still on my Apple Music account.

In theory as long as I can still play my music fine, but I shouldn't have to manually double click all 12,896 tracks in my iTunes and click Cancel just so the software can remember it can still actually play them. There must be a mass-library fix for this. Any suggestions?