Music purchased on iPhone not showing up on Mac

Hi,

I do not subscribe to Apple Music.

I have a couple of albums purchased on my iPhone that are no longer available to purchase or download from the iTunes Store.

I can play these albums on my iPhone no worries and they are downloaded on my phone. But on my Mac, these albums will not play. They do display in my library, but when I go to play them from my Mac - the song won't play + I get a little ! error that says "The song “...name...” could not be used because the original file could not be found. Would you like to locate it? I have checked my media library on my Mac and it doesn't look I have ever downloaded these albums on my Mac, so there is no original file to connect (and re-downloading is not an option).

I found a couple of articles on here about iTunes for Mac that explain how to sync music from my Mac to my iPhone. My question is this: If I follow those instructions to sync music from my Mac to my iPhone, will these whole playable albums sync from my iPhone to my Mac? I'm not game to try in case I lose the files from my iPhone.

Thanks