Is it legal for my representative social security payee to have a bank account in my name that I don't have access to?

The title of the account would be a combination of your names, like their-name for your-name.  Perfectly legal.  And unless there are huge fees, an excellent way to ensure that they spend the money as directed by law.

More than eight million people, who get monthly Social. Security or ... Remember, the law requires representative payees to use the benefits properly. ... If you don't get medical treatment for the child, Social ... Bonds or an interest-paying bank account, insured under either federal or ... (Your name), representative payee for.

That's the entire point of having a payee.

You are deemed incapable of handling your own money, so someone has to do it for you.  The money is yours for your care.  But you can't directly access the money.