You do have it on your computer: your server.

So when I listen to music locally, I just open clementine and it knows all the music on my server, and plays from the network share.

When I go out, I stream to my phone from the same server. How often is your phone NOT on the internet? I mean its a phone right?

Here I am about 5,000 miles away from my server, and I still stream to myself, to the laptop I am working on. Should I load up my phone, laptop, tablet, and steamdeck prior to leaving the house, or just know I can stream anything from my 6TB collection (seems big, but I only do FLACS) at any time?

If you DO feel you need to have some offline, it still makes sense to have a server. Just use syncthing or any decent music program to bring over what you want before you leave the internet.