We have plenty of examples of "bad news" that people accepted with sufficient proof. Like the link between cigarettes and lung disease, for example. It's true that if we're delivering bad news (i.e. news folk don't want to hear), then the evidence needs to be solid. But it's not true that good evidence fails to persuade, given time and the opportunity to consider it.
People may accept the link between cigarettes and lung disease but they don't quit. Accepting the evidence that Jesus is the Messiah is not like accepting the evidence that DNA is arranged in a double helix; whether that is true or not will not change anything in my life. But to accept that Jesus is messiah means I must change the way I live.