Yes, if your mobile app is a PWA, you would definitely need to transcode most videos. I'm not sure of how you did that so far, but in my experience, all web browsers support some specific codecs, and nothing else. Same for container formats. For example I haven't been able to open a Matroska (MKV) video with either Firefox or Chrome. And if those browsers have their limitations, I can only assume iOS Safari is even more strict with what it can consume!

I think your idea was the safest one: transcode the file in the main PC, then send it out to a <video> tag in the phone; I'm surprised that didn't work. What protocol is the transfer done with? Maybe iOS Safari doesn't support it, or if it's plain and simple HTTP, it might silently fail if the source is not HTTPS with a proper certificate.

You could try WebRTC. Safari supports it, and it's well tested at this point. It's more complex, having to send SDP messages back and forth, and having to care about browser limitations (e.g. videos cannot autoplay if they have audio, things like that). If you decide to go that route, this project might be just what you needed for this task ;-)