They aren't "persistently wrong" -- it's just that, by definition, only the times they *are* wrong are memorable -- and then that memory is often distorted anyway to make the difference even more dramatic than it actually was. Same with telling the forecast for the day "by just looking at the sky in the morning". I mean, really? Depends on how precise you define "forecasting the weather" I suppose. I can look out right now and say that it's going to be cloudy for a while today but couldn't be any more accurate than that -- will it rain? If so when, and for how long, over how wide an area? When, if at all today, will the clouds disappear? I reckon that the average man can answer exactly none of those questions with any real confidence at all. But, of course, if you did, and happened to be right for a change, you would again forget the times you were wrong.
Finally -- the source for the forecast from your smart phones and from the TV is ultimately pretty much the same; either from Met Office Supercomputer calculations, or from some other supercomputer calculations elsewhere. Doesn't change much.