I did a factory reset of iOS 10 and set it up as a new phone. During setup I said "No" to enabling iCloud Keychain. Afterwards I reset iCloud Keychain on my Mac by removing the security code > uncheck iCloud Keychain > Restart > Enabling it again > Reset iCloud Keychain. After that ordeal was behind me, I enabled iCloud Keychain on the iPhone.
Everything started working on my iPhone's Home app at that point, but my partner and roommate got kicked out. I tried inviting them through the Home app, but it isn't being accepted by the Phillips Hue app. It gives out an error that they can't connect because another iPhone is already in use.
So at this point, no matter what we do, only one iPhone can connect through the Home app and Hue Bridge at a time. Frustrating as ****. But at least no one is stuck at a loading screen.
Remember when Apple stood for simplicity?