iTunes cloud library not being synced on a new Mac

I have a new MacBook Pro and I lost all my data on previous one, unfortunately the backup was not done :(.

I have my iPhone and iPad backed up on icloud. I have Apple Music. Everywhere I ticked "Sync Library".

I wanted to sync iTunes with Mac to have all the music from iPhone also on Mac.

On Mac iTunes I see only titles of songs, but I cannot play them (they are grey). It is the same on web cloud account (music apple library).

If I connect iPhone to Mac with cable I am afraid data on iPhone will be lost, usually music from Mac was put to the iPhone.

How can I put my iTunes library from iPhone to Mac (to have it also physically on Mac)?

Thank you!

Aleksandra