Interesting. Thanks for the detailed write-up.
As an alternative you can in fact do the same thing with the iCloud account as opposed to an outlook account.
Here's what I do:
I have a rented personal email service under my own domain. Unfortunately, it is POP3 only and doesn't support push. I could add Exchange services to it but that's not that cheap and I don't feel like switching providers for a variety of reasons.
So what I do is essentially emulate what I've been doing for years with my Blackberry (where this was easy to setup and actually intended to be used this way). I forward all incoming messages to my iCloud address which immediately pushes them to my iPhone.
Problem is that iCloud (officially) doesn't allow the "sent from" address to be changed. So instead I'm setting up the iCloud account manually with the generic server settings I found somewhere and then use my own server as outgoing.
I'm pretty sure this type of setup would work with a Gmail account as well for those who prefer that.
Why in this day and age we have to jump through hoops just to get push mail from our preferred email service in an email app of our choice is beyond me. Like I said, the old Blackberry BIS and BES services were the easiest to use and most reliable for stuff like this. What a shame.
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Does that support push mail? I thought only the generic Apple mail app handles push mail? (on those days where it's actually working without hiccups)