This is simply the way Apple has setup Notes sync over iCloud. It syncs using the IMAP protocol, so probably in an effort to not confuse some customers, it has chosen to make an iCloud email account mandatory as the iCloud email using a variation of IMAP.
There may also be self-serving reasons too, such as being able to inflate their actual usage numbers, but I do believe it's more to do with causing less confusion with some customers.
However, if you go to Settings --> Accounts on your Mac, you should be able to add Notes sync to your Gmail account that you mentioned you owned. Although this may not work as "quickly" as iCloud would and may not have all the features that could be added at a later date by Apple.