If I believe the Clinton's are pedophiles and murderers and are part of a ring of like minded people, and I'm inundated with information from people and organizations which support this (or at least carefully don't refute it), then when I'm presented with information about a pizza parlor that is supposedly holding children in the basement, is that consistent with my beliefs?
I think what you're presenting is just what everyone already does. Instead of assessing thi gs based on how well they fit our beliefs, we should assess them based on a consistent objective standard, and then alter our beliefs if it meets that standard but conflicts with our beliefs.
This may in fact be what you belief, because you belive in facts and the importance of the truth. The problem is that you get wildly different results when someone that values different things applies the same system.
This is analog to growing the tree, the page talks about cutting it down.
One could give many examples but the good ones are unlikely to resonate with others.
To give a poor one. There was a time when I understood human decision making as a hierarchy of people in increasingly greater positions of power with access to better information and to people with greater skill. Then one day it struck me how they too are just going though the motions with their freedom for creativity limited to a single potentially career ending move. The machine happily grinds on without anyone behind the wheel.
Stuff in life is complex, people are assholes, but even assholes have good ideas sometimes. I recommend listening to everyone who speaks for themselves in good faith. Anyone can cook!