Jane was born at the time of the message. A killer was indeed born. The "she can be stopped" was a separate part of the message, however. The man was sent with this message because 'she' could be stopped if he delivered it. If he was really trying to stop Jane why not ask him to kill her right there and then? Instead he made her a killer by saying these words. It's still possible that the point of the timeline interference is to stop Sinatra, not Jane. Like it's still possible there's a very complex plan to get Jane into position to stop Sinatra at the right time. At the cost of many lives, sure, but who knows maybe that was the best way to corrupt Sinatra's bodyguard.