Remove iCloud email address assiociated with my Apple ID.

Apple support says that I cannot delete it

This is part of the baggage of history. My primary email is one of the @mac.com addresses from the time of .Mac (2002-2008). During each of the migrations which followed - MobileMe (2008-2011) and finally iCloud (from 2011) we also gained the @me.com and @icloud.com. This is just a part of the evolution of Apple's cloud service.

Because it is built this way you can't abandon the origins of your account. You will find that you don't have to use the @me.com address for emails and can instead use the @icloud.com address if you wish, although the @me.com address will remain your account name.

If you wanted something different for email you also have the option of creating up to five aliases (at icloud.com) all of which will point to your iCloud In Box, and Apple's new Hide My Email service will permit some further options for anonymous addresses for filling in web forms and so on.

You don't have to have an email address with them to have your Apple ID.

That's true. Those who sign up to iCloud and use an alternative such as @gmail.com can use it quite happily, with the exception that they don't have iCloud email, obviously because it's a Gmail in that example and a different service provider. So they have every other service but not the email. When you discussed this with Apple Support did they definitely say you can't change your ID to another email address form?

The support document Change your Apple ID suggests that you can do this, and the graphic on the page even has an iCloud email as the account name being changed. It's an easy process, and I've done it myself for somebody although that was from one non-iCloud address to another non-iCloud address. But I've no experience of what happens if you change from @iCloud to @whatever so I can't offer any advice I'm afraid.