auto-saving video chat on iPhone

Hi, I'm using my iPhone linked to a video projector in order to have max size when video chatting with element. I'm on iOS 14. Now this setup doesn't allow me to record the video chat, because iOS apparently refuses to do so when in airplay mode for some reason. So it would be nice to have the option to have element automatically save video chats so I can still project the chat on the big screen. This should also be possible on the desktop versions really...