The loss of privacy in this case may not be worth using a public ledger.
In general though, I think it would be an interesting experiment if individuals can hold central bank money directly instead of going through commercial banks.
I think there are plenty of ways that privacy could be regained... layer 1 settlements, use of an anonymous protocol like MimbleWimble, etc. It wouldn't necessarily need to be Bitcoin as it's implemented today.
But I agree... that experiment would be quite fascinating. It might not even be that big of a change... most people would probably continue to entrust a bank with their funds, and banks could potentially continue to operate a fractional reserve system.
In general though, I think it would be an interesting experiment if individuals can hold central bank money directly instead of going through commercial banks.