FYI Personal experience on Social Security record update after naturalization:
1. I became a US citizen in 1987 and nobody told me I needed to update my record before the SSA to inform them about my new status, so when I decided to retire from military and civil service in 2021, I had to prove my US citizenship before the SSA and it was very simple, I received a letter with an appointment asking me to come in person with the US naturalizatiion certificate, they made a copy of it and made few entries in their system and that was it, I was not asked for a US passport.
2. My wife became a US Citizen last August 01, she updated her driver's license record, applied and received her US passport, updated her new status with banks and brokers, received her voter's registration and today, she had her appointment with the SSA requested 3 weeks ago.
Now days, there are no walk-ins at the SSA, its by appointment only and one of the ways to obtain one is through their website using your online account, which ask you a bunch of questions so they can decide if the appoinment is necessary, well the only option in this case is to present documentation in person, and that quaifies you to finally obtain a freaking appoinment.
When we showed up today, the lady who tended to us said we applied incorrectly and gave us a lecture stating there was another option to update the immigration status, etc. (believe me, there is no separate option for that) and directed us to go to another window to use a terminal and complete a very long form which asked for a PASSPORT NUMBER, date issued and expiration date. That is the big difference between my process in 2021 and today with my wife's.
After we completed that application we went back to her window, then she asked for her Naturalization Certificate and her driver's license, then she made a couple of entries and printed a letter stating that in 2 weeks my wife will receive a new SS card with the same number.
So, in SUMMARY, seems that in order to update your SS record, besides the Naturalization Certificate, a US Passport is needed also. (We tried to continue the form not providing a US Passport info and the system will not allow us to continue.).