I do recognise that a lot of it is probably just playing up for the joke, but I assume it has to come from somewhere. I regularly see posts where basically Person A sends an odd image/message, then follows up with “wrong number, sorry” and Person B responds with “wait stay”.
Do Americans, as a rule, not save phone numbers? Or is it purely a bit that has become a trope?
No idea what you are talking about. Have you ever considered the idea that maybe you spend entirely way too much time on the internet?