Why can't I see my entire iTunes library on my iPhone?

I'm logged in with the same Apple ID on my iPhone and my computer

There are too many songs to physically download onto my iPhone, but I still want to see all of them, and play them when I have Internet. For now I am using the Google music app, no subscription fee, and it seems to be working fine . But the reason I bought an Apple device was for access to my iTunes library.

Homesharing works sometimes. But only on my home network.