No, I really can’t understand the mindset. Especially not in the face of the constant undermining of trust by certain elements of society, including when they’re in government. We didn’t just arrive here for no reason. The same people who have eroded the trustworthiness of government and authority (on purpose, see Reagan) over decades are the ones who now exploit the results of their actions, for their own gain.
If, in your scenario, group B was on the level, it would be a different story. But they aren’t. If A oversold their claim, B would have massively oversold theirs. And that was easy to prove and has been proven. B also just didn’t oversell their own claim, they also exaggerated the claim that they refuted to something that, in this form, was never said - standard MO.
There is no trick to this. Being factual and getting people to believe you is much harder than telling an easy but good-sounding lie and getting people to believ you.