When central bank coins will be out the main FUD central banks will have toward Bitcoin will be lack of privacy. People will get very sceptical of using Bitcoin and give their information's for everyone to see compared to have same privacy as they have right now with central bank coins.
There's definitely more privacy with BTC than there'll be with CBDCs. What these guys'll do is they'll make it so that you'll probably have some kind of imaginary privacy, something similar to the COVID-tracing apps. You theoretically have privacy since the data is exchanged randomly, yet it's not that hard to link a device to their owner or two devices to each of their owners.
We are pseudonymous. CBDCs will be fully open to the authorities, something very similar to the banking system we have today. What I'm more worried about is that Bitcoin doesn't have any privacy besides coin control, mixers and CoinJoins, out of which the first is a quite difficult task if you want some persistent privacy while the other two are seen as coin-stainers. If that many people want privacy for a digital coin, then I'm afraid neither BTC or CBDCs are the proper answer.