Turning off the cloud service just does the syncing thing right? What about turning off the Apple Music service so that the only thing visible is your local content? That's what pisses me off the most.
I've only ever used Apple Music with local content. On iOS the only indication to me that non-local content is even possible is the radio tab at the bottom of the screen.
On MacOS I think it opens to the online home page, but I use it so infrequently I'm not sure. I pretty much only use it to buy music from iTunes.
I've recently tried syncing local content to my phone, but to find that content on the phone is difficult. The phone really wants to show me Apple Music stuff. I have to Library->Downloaded->Songs. Going to Artists or Albums just shows me "Download Music to Listen Offline". I really just don't want to spend the time I previously spent on my iTunes library all over again. I was really just trying to quick&dirty add content. They've made this unnecessarily difficult and I despise them for it.
Just go to the Library tab in the phone app. That's where your own stuff is located.
I have over 8000 songs synced to my phone, 100% from local files on my computer.
I never activate the "library sync" BS that Apple tries to force on you, because historically it has replaced your copies with incorrect or "remastered" (AKA dynamically compressed to hell) versions from Apple servers.
I've even caught it switching "library sync" on without permission during an update.
I don't know what you're doing differently then. When I open the Music App, it goes straight to my Library. The only items in there are songs I've synced to my phone. I've never seen anything else or known it to behave differently.
Thought if I remember correctly, search was still showing it, which was a little annoying. But it depends on how much you search.
I'm currently using the service right now, so I can't really check if anything has changed since last year when I was doing it.