Has this ever actually been a problem?

Do you know how many utterly inane creationist ideas have been shot down by people who care about science? By taking the utterly inane idea and explaining why it's utterly inane, making a total fool out of the person who pushed it (in everyone's eyes but their own, most of the time). This didn't make creationism part of "legitimate discourse"--people already wanted it for ideological reasons, but everyone viewing the "debate" (and some of the people on the side of idiocy) could see that the ideas were really stupid. Sometimes the arguments were too long to fit in a tweet...but no problem! Well-referenced and well-written responses to all the common claims were put up at, say, https://www.talkorigins.org (originally talk.origins usenet) and...hey! They're still there! Need to debunk "polystrate tree fossils"? It's one link away.

We don't hear much about polystrate tree fossils any longer.

It's only when you don't have good arguments that you need to fear "making them part of the legitimate discourse". Because if you don't actually have any arguments--maybe you never had any, or maybe you forgot them--then of course some people will think, "Well, that other idea sounds plausible too. Maybe that's actually right...let's try it on for size!"

So, if your only argument against Nazis is "OMG, like HITLER?!!" then of course any need to argue against Naziism--which should be child's play, given how it's based on factually incorrect assumptions, does have the case study of "OMG Hitler" as an example of how it can go horribly wrong, and is predicated on an indefensible moral standard (i.e. it clashes grotesquely with most moral intuitions of most people, and it is only by actually denying most of Naziism that people can even start to stomach it themselves until they grow accustomed to it)--suddenly becomes a fearful endeavor. What if we don't dismiss them flagrantly enough? What if we don't shout them down belligerently enough? What if we don't cancel them viciously enough (and their friends too, if their friends don't cancel them)? What if, despite all our social cues to the contrary, they start to believe?! What ever shall we do?!

What indeed? Maybe use some arguments. Real ones, not just social pressure.

This is why I think it's really not up to conservatives. I mean, it would be great if they could police the utter rubbish that their extreme wing puts out and which a lot of them end up parroting because they don't want to lose face with the loud ones. It would be great if the left could do that too with their extreme wing.

But honestly, the conservatives only have any power in a system like ours because they appeal to enough of the moderates, centrists, and undecideds. Despite all the effort put into hyperpolarizing things for the past three decades or so (let's use Gingrich to mark the start?), there's still a lot of middle. The middle doesn't like extremism. Indeed, the 2022 midterms gave the Republicans a stark lesson in exactly that--the extreme candidates did poorly, on average, as did the Big Lie proponents, as did those who hewed closely to Trump.

Arguments have room to work on the middle, because the middle isn't captured by ideology (or they would be part of one or the other polarized side).