TL;DR: If you change your gamertag on the Microsoft site then, at least under some conditions, the iOS app keeps trying to connect to the old one and won't launch anymore.

So I ran into precisely the problem hit by joshkelsey-ross on Microsoft Community last fall:

My son has had minecraft on his ipad for a while now, and i have just got it for my iphone so that we can play together. It has been working fine until today when i created a microsoft account for him and set it up as a child account with myself as the parent. Now when he tries to load minecraft, the mojang logo comes up, it tries to load and then it closes itself. 

I noticed that the app was almost loading before crashing, getting through most of the startup screen.  After restarting the iPad and letting it sit for a while before attempting to launch the app again, it got all the way to the welcome screen and displayed a message something like "now logging into your Microsoft account" before the app died.

I was preparing to uninstall and reinstall the app to try to get it working at all, worried that all my son's levels would be lost in the process.  So first I connected to iTunes to backup the iPad and any accessible Minecraft files in the hope I might be able to restore his games somehow if that happened.

On a whim I looked inside XBLStoage.json at the root of the file structure.  I noticed that in two places his gamertag was still recorded as the default one that the Microsoft site assigned to him when I first created his account – he had just changed his gamertag before the app started crashing.

I hand-edited the two incorrect gamertags to his new gamertag and put it back on the iPad, and the iPad app resumed launching.