Since the policies of 2008, bank IOUs and central bank IOUs are basically the same since the central bank backs up all the other banks. 100 dollars held at Bank of America as "this is how much the bank owes me" is functionally the same as "this is how much the fed owes me"
In theory, private banks would've created a diverse ecosystem where bad or incompetent banks would get washed out leading to a stronger and more efficient private banking sector. In practice, they all needed to be saved. The resulting rules like lending standards coupled with existing regulation like KYC means that banks already act like arms of the central bank.
It's possible a CBDC just cuts out the middlemen so there could be benefits like lower interest rates on loans. There is concern with privacy and centralization of course, but that's not specific to the CBDC idea or tech and has already existed in a form for some time (e.g. canada trucking bank freezes)