No. You have access to this account, so you don't need probate to get access to the money. If you are able to do all that is necessary so the legal heirs get their inheritance without probate, there's no need to go to court to get it. I'm British, you're American, but this is unlikely to be much different. Probate is more about giving the executors legal authority to get their hands on the assets and/or legal authority to transfer ownership than anything. It has no relevance to who actually gets anything.
You mention insurance policies in a comment - if the insurance companies will pay out without seeing probate, then you won't need it.
This rather reminds me of my grandad. Everything he had was in joint names with Grandma except for a few very small sums that the financial institutions concerned were happy to let out just on seeing his death certificate. His will said everything goes to her and my Dad is executor, so all Dad had to do was clear up those very small sums and claim on the insurance policies. None of them asked to see a grant of probate, so it was all done without it. Of course when Grandma died, all her money was in her own name and we had to get probate.