I believe that probate is different in each state. My dad passed away last year and I was the 100% beneficiary to everything. In Florida, I could sell the house but only to a cash buyer, if I wanted to sell it before it finished probate. In Florida, only the property goes to probate. Before I could put it on the market, I went to an attorney to start the procesS. I actually sat down with the attorney within a few days of my dad’s death. I had forms to fill out and couldn’t do anything until I received the death certificate, obviously. It took 10 days just to get the death certificate. If I sold the house to a buyer with a mortgage, I’d have to wait until probate was done. I was able to get a cash buyer and the house was sold 2 months after his death. However, probate wasn’t complete until October 2023, he passed away 1/1/23.
As far as needing my actual social security card, I do not carry it with me. I have never needed the actual card but believe I did for my drivers license (NJ) as one of my 6 points. I do have a passport, but think I also used my social security card.
edited to add that I needed the actual card when I change my name when I got married.