You can download SS-5, fill it and bring it with you to SSA to save time. I'm not sure if it can be done online. When I removed restrictions on mine, SSA wasn't open and I did everything online / by post. I had to mail my actual GC to them!!! I'd think if it's still allowed, you'd me mailing naturalization certificate. I wouldn't personally take the risk. I had no choice during COVID, but now I'd walk there in person.
No it wasn't mandatory. But it allowed you to only show SS card and DL to employers and not GC (plus any extension letters)