The gov doesn't need your social to find your info. Everything you put on the 4473 gives them more than enough info.
These people can see it all.
I said it before, when my mother in law was traveling to the U.S for the first time, TSA stopped her because they were being d*cks (just long enough for her to miss her flight then let her through). She said her daughter's info, and MY information, was on their screen (we weren't married but lived together). Everything was there.
She couldn't read it all, but I wouldn't be surprised if it also said I owned guns.
And this was some random TSA person, nothing top secret. I managed to find the right phone number and called them while my MIL was there to ask "WTF". It was funny, she said they answered the phone near her and she could hear me.
... the idiots asked me how I found the number, all I said was "that is my job". That felt gangster.