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.