When you enable Azure Active Directory (AAD) authentication for Azure SQL, you are not required to enable public access for Azure SQL Networking. AAD authentication can work with both public and private network configurations.
The issue you're facing with the private endpoint might not be related to AAD authentication but rather to your networking setup. When using a private endpoint, you need to ensure that the virtual network (VNet) and subnet configurations are properly set up to allow traffic between your client machine and the Azure SQL server.