Take this advice with a grain of salt as 1) I’m subscribed to Apple Music and 2) I’m on iOS 14 so I’m not sure if the menu options change under both of those conditions.
In Settings > Music, do you have the options to Sync Library and Optimize Storage (under the Downloads subsection)? Not sure if it makes a difference but if you see those AND they are presently both turned on, try turning them off.
Not sure if it’ll help but they’re the only settings I could foresee causing that problem for you. Otherwise, I genuinely think it might be a bug because I only this past 6 months started subscribing to Apple Music and had previously done the same as you—synced my own personal music library to my iPhone without the unfortunate issue you’re experiencing.