I am somewhat confused by your reply. I don't know what you mean by the local calendar that only exists on the phone. Under Settings>Mail, Contacts, Calendars>iCloud>, I have mail, contacts, calendars, reminders, bookmarks, notes, all set to "on" under my iCloud account. I have two other accounts for which I receive mail; only mail is "on" in both those.
I tried the trick on my iMac of System Preferences>iCloud>unchecking Calendar>delete from Mac>waiting a couple of minutes and then rechecking. That seemed to work for a brief time but now I am back to not getting appointments on my Mac that I make on the phone. Just now I tried a test appointmet from my iPad to my Mac and that worked fine. It did not appear on the iPhone, however. Should I just wipe my phone and start all over?