A few ways I've had success finding these kind of tracks:

Checking other regions of apple music. You can do this with a vpn, but there are websites which tell you which regions a track is available in. For some reason India has been the last remaining hold out before.

With a bit of work, you can find the preview file on spotify, even if it's been removed. You need to find the song id, which you can extract from spotify web if you manage to get to the "this song isn't available" page, and then there are websites that help get more info on these tracks.

Additionally, you can reverse image search the album cover, sometimes this throws up things.

Sometimes these tracks are library music, and a client has picked the "I want to by exclusive rights to this song" option, which removes it from other circulation. Related to this, it's not unknown for the same library tracks to appear under different artists names and song names, as either the writer tries to get back to the top of searches, or unscrupulous people steal the track and submit it under their name. Trying shazam again, and try the other recognition services, like soundhound, google's one, etc. They sometimes have other listings of a song.

You can also contact the label, or see if you can find any of the personnel who worked on the track. I've watched songs I've produced be pulled from distribution by the artist's label before, to allow them pitch the artist in a different image. It's infuriating as it effectively means my credit that I slaved for countless hours to achieve is gone, and I also really feel for the fans. I would definitely (and have done) given out copies of the track if I still have them.