iCloud features requiring use of @icloud.com account versus actual account name
I had a similar issue and eventually got sick of dealing with the issues you have noticed and gave in and used the icloud account. Calendar, app store, reminders, notes, etc. It all works better with the icloud account. I never use the email address associated with it (and I've had one since the @me.com days).

Yes it is annoying but Apple's built-in apps work much better with Apple's accounts. You can spend hours of your time troubleshooting or moving over to dedicated 3rd party apps or just "grin and bear it."

I know that doesn't solve your problem, per se. And there may be folks who have found acceptable workarounds (if not here try macintouch.com) who can offer a better solution but I found that not fighting city hall ended up working for me rather well after all... YMMV...