Why do i need a new email account to sync my notes in iCloud?
iOS Notes syncing works through the IMAP protocol, which in iCloud is integrated into the email service. Although Apple could technically choose to allow for notes syncing without explicitly requiring an email address, they have chosen to make it a part of the email, almost like Notes is its own email folder.

You should also be able to sync your Notes using your Gmail account since it is also based on the IMAP protocol: