Ryan,
After speaking with folks for Esri Technical Support, we have tested and reproduced the issue you presented. and I think we figured it out. In order to share the link for iOS device, the AppStudio OTA website, AppStudio OTA, and the "Share" button needs to be opened from another iOS device for the bit.ly link to download and install the IPA file.
If the "Share" button is clicked from a non-iOS device, then the link will only download the IPA without installing it.
Thank you,
Erwin