Something not being provable underlies the definition of the term theory, Google this phrase, "is it required for a theory to have proof" to understand.
"It does not matter if they end up being right"
This is not a logical stance.
And, yes, it does not matter if they end up being right, the same way it does not matter if a broken clock end up being right: it is still a very very unethical and unfair way of managing a society. I can throw knives from the top of my building, and sometimes I will hit someone that deserve it. According to your logic, it's fine.
On top of that, if you learn they end up being right, it means that someone else came up with the proof, demonstrating that they were utterly useless: their "belief" did not change anything, because the proof would have appeared anyway. All they have done is being counter-productive and delayed the discovery of the proof (and sometimes ruining it totally).
It's a bit like bad cops that "get the bad guy" by breaking the rules and end up with the bad guy being released due to breach of procedure. It's just totally stupid: if they believe there is a problem, they investigate and bring proofs upfront, not start accusing without proofs.