@pjh As you have mentioned that this is a concern on the iOS app specifically, but also on wifi as well (as you said mobile data seems to work on the iOS app), my assumption is that this may be with the wifi network connection on your phone, and not the network itself. What wifi network are you connected to on your phone? I recommend contacting your ISP in this event, as there may be a port or protocol that needs to be opened to allow the wifi to load a Live View (uploading live streaming is how you can generalize this for the ISP). It is odd since other devices on the same network seem to connect, but there may be something specifically with the iOS phone’s wifi connectivity that needs to be addressed.
If you find no relief in this, I am curious as to what iOS software you are on, as well as what iPhone you are on. I myself am on an iPhone 6s with iOS 13.3.1, and have no concerns with Live View whatsoever, even on this “older model” iOS device.