When you say "demonstrably untrue", who determines such things? It is usually not the case that one side is entirely and wholly untrue. Losing a court case or having one be dismissed on procedural grounds does not mean the defense or prosecution had no merit whatsoever with their claims.
What is the process for letting people make up their own minds on conflicting evidence? I sense the danger is letting, or even demanding, that private companies make these judgments and filter dissent away from the masses under the guise of safety. I also sense the people calling for such things understand that their own ideological peers are the ones who would be in control of such algorithms.
Rather, it's a casino. And that's not value-neutral, freedom-neutral. It's pretty antithetical to freedom, as you can see from all the people destroying their lives at the gambling machines.
What determines truth is empirical evidence. If something has no empirical evidence, it is untrue. If something has empirical evidence, it is true. If there is conflicting evidence, then some of that evidence is invalid and it must be re-analyzed using math, existing knowledge from provable things, and formal logic. After doing so, you will either determine which of those things is true, or arrive at the conclusion that there isn't enough evidence either way and stop after saying "I don't know" rather than deciding which version you prefer.
It's not about appeals to authority. Expertise is about people who have more practice at finding, testing, and analyzing the evidence in their field than random Joe Schmo on the street; and it's about nothing more than that.
We should not let people "make up their own minds" on conflicting evidence -- which is another way of saying "let people make up their own reality and expect to live in it" -- we should encourage everyone to stop at "I don't know" when they aren't sure where the evidence actually leads, and defer to people who can follow the evidence, if such a person exists. And if no such person exists, then we as a species should all stop at "I don't know (yet)".
Do you believe in objective reality?