As you point out, you can’t recreate an experiment to prove that Christ existed. History works differently than other sciences, like biology. You are always dealing with probabilities. So, authority takes on a different meaning. The fallacies you point out are valid, but should not be applied in the same way they would be applied to biology. In biology, a grad student’s experiment could trump a tenured professor’s opinion. There is no such trump card in history. Either we do all the work ourselves to convince only our self, or we rely on the consensus of opinion. When doing that, I also look at whether or not other historians acknowledge someone’s opinion and vice versa. In Murdoch’s case, I also note that rarely mentions other historians or critiques their work or explains why her opinion differs. She only talks about the ones she uses and the parts of their work that support her POV.
Also, do they have an agenda? You mention theologians do, but fail to note that non-Christian historians also agree that it is most likely that Jesus existed.