Is there a way for me to just share Apple Music and nothing else?

Yes, keep reading last paragraph listed in previous answer. It states

"Not all content is eligible for Family Sharing. ."Content can be hidden by family members;" hidden content is not available for download."

Yes. Instead of setting up Family Sharing, then restricting it, you can subscribe to an Apple Music Family Plan.

Get an Apple Music Family Membership - Apple Support

Family purchases and payments - Apple Support Toward the end of the article on family sharing it states you can share only the purchases you want to share, buy I think that's but category... so you can share music, but not photos, or games but not music. Unsure if you can mark specific items as not shared. Probably have to go thru the setup process to get that far.