Nope, I personally started a new job during ROC. It took them a while to figure out how to use the extension letter, but it worked fine. This was during covid so the Social Security offices were closed, but when they opened I got an unrestricted social security card. The only annoying thing is that if you are in the middle of ROC when you start, they can ask for reverification once your extension letter expires, so I'd definitely go the unrestricted ss card route.