Organizing and accessing contact information in Outlook.com
It sounds to me like there are too many programs involved in yours. I just use the native iPhone apps for phone, calendar, notes and it all works with my Outlook on my laptop. Originally the phone didn't sync contacts made on the phone. Now it just magically does. I suspect because of an update to either IOS or Outlook. Both have happened since I started this thread.
I have an iPhone 11 Pro, IOS 13.5.1, Outlook 365 on my laptop with all the latest updates.
Maybe Outlook for IOS is making it not work. I found that using iTunes for syncing isn't good because you don't get instantaneous syncs. And I can't stand iTunes as an interface.
Have you tried the native iPhone apps? Do they not work for you? Do you need to use some form of Outlook for iPhones? If you can just go back to the simple versions and try that, it works. (Although could this be an older phone issue?)