I'd handle that balance by dropping the "not legitimizing their opinions" part completely.

There's a lot to unpack here, but briefly, the idea of "legitimizing" an opinion suggests that there is some group of people who have the authority to decide what opinions are okay to hold and what aren't. The problem is, a) there is no dispassionate authority you can refer to when we're talking about political issues; everyone brings their own biases whether consciously or not, and b) people who already disagree will not cooperate with that authority voluntarily, and c) the authority has no actual power in a democratic polity anyway. You can't arrest the other side, you can't put them in camps, you can't fine them. All you can do is yell insults at them, and if they don't care what you think, you are effectively powerless -- and obviously powerless to anyone watching.

Your only option in a democratic society is to have faith in the truth, plainly and calmly and repeatedly and respectfully spoken. When the other side comes at you with illogical nonsense, don't get angry; look at it as an opportunity to educate the onlookers, an opportunity that the other side has generously given you. And take that opportunity as often as they foolishly provide it to you.

And yes, one can be cynical and believe that the truth doesn't always win out. But suppression of lies either ends up in the fascist state you're trying to prevent, or eventually collapses in on itself and creates an even worse reaction. In an open society, the fascists are a couple of losers in basements complaining to themselves while the world moves past them. In an increasingly closed society such as the one we live in now, they become the brave rebels fighting the Establishment. Don't let them be the brave rebels.

Thanks for the opportunity to express these thoughts.

Very well said. I think you're spot on that these folks aren't going to back down, but at least you can use the opportunity to explain the facts to other people who might be more willing o hear them. That's something we're missing during these discussions, we put a lot of emphasis on hearing both sides of the story, but not a lot on making sure everyone is aware of the facts, even if it hurts the rhetoric of one side of the argument. Fact checking during debates or roundtable discussions would go a long way towards rectifying that.

Thanks for the conversation, I do see your point and I agree. It's a no-win (or at least a only-partial-win) situation, but stating the facts and moving on is probably the most effective route, with the hope that the truth wins in the end.

They're just so damn annoying though!

One way to make those interactions less annoying is to, perhaps temporarily, put aside your goal of getting them to agree with you, and choose another more achievable goal. Here are a few you could try:

Have a cordial interaction with someone who sees things differently

Learn from this person about one of the things they are an expert on

Understand more deeply the pseudo-logic of their mindset and where it comes from

All of those goals are realistically achievable which should make them less frustrating than trying to win. And as a side benefit if you succeed at any of the above, you should be in a better position to bring them around should you go back to trying to do that.