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Lumifer comments on Rebutting radical scientific skepticism - Less Wrong
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Otherwise known as the Typical Mind Fallacy :-)
But yes, you are correct, as long as your main criterion is something like "compelling at an emotional level", you should expect that different people understand it very differently.
This actually brings out something I had never thought about before. When I am reading or reviewing papers professionally, mostly the dispute between reviewers is about how interesting the topic is, not about whether the evidence is convincing. Likewise my impression about the history of physics is that mostly the professionals were in agreement about what would constitute evidence.
So it's striking that when I put aside my "working computer scientist" hat and put on my "amateur natural scientist" hat, suddenly that consensus goes away and everybody disagrees about what's convincing.
That may be a peculiarity of physics (and math). Compare that to biology and medicine, not to mention social sciences.
Well, of course, because "compelling at an emotional level" isn't really about evidence. Cute puppies are compelling at an emotional level.
You're basically talking about getting a "proper" gut feeling, and that is very idiosyncratic.
Your observation of little conflict about whether the evidence is convincing could be explained by a consensus about whether it is convincing, but it could also be explained as low priority. That is my experience in math.