For one, they lie. They lie, lie, lie. Nazi’s lie more naturally than they breath. What they want more than anything is a platform to spread those lies to people who are vulnerable to those lies. Hitler didn’t seize power, he was elected into it.
They want a platform, any platform, for mainly 2 reasons, neither of which are for the sake of logical debate or upholding freedom of speech:
to spread their propaganda, which again is full of manipulative lies that feed on pre-existing biases. People don’t become radicalized from nothing, they’re fed a specific narrative that leads them down a road of reasoning that feels, to those people, like it makes sense, until they’re in too deep to be willing to consider that they’re wrong.
to UNIFY disparate groups that share their views or are at least willing to brush elbows with the nazi party to achieve their ends. This unification of disparate groups can lend them a lot more strength and influence than they’d otherwise have if they were kept separate. They want a platform that let’s them, essentially, stand up on stage with a microphone going “alright, hands up if you share some of these views and are willing to go along with the rest as long as you get what you want in the end”. Now they won’t state the worst of their views outright, they’ll just couch them in nicer terms, but people who share them will recognize them. And even those who don’t agree with the extreme versions of those views are just targets ripe for radicalization.
So then not only do they have a platform, but they have larger numbers flocking to this platform, which gives their voices more weight, and makes them feel more supported in their views. It makes them seem more reasonable, more legitimate. It makes them more appealing. They don’t just stand up at the podium and say “gas the Jews”, because they’re not being honest. They say stuff like “I just think we should be concerned about how many Jewish people are in positions of power, using that power to their own ends. I just think it’s strange, that’s all. That doesn’t mean I’m advocating violence. I’m just asking questions, I just have concerns.” and they’re not advocating violence technically, yet. Not in public at least. And if you say" hey that sounds awfully anti-Semitic" they deflect. “it’s just an observation of a fact. Apparently facts are anti-Semitic now”. They have any number of excuses ready for why their views don’t actually count as being anti-Semitic, and if you say otherwise, we’ll, you’re just trying to silence them. Because that’s the thing, once they’ve established themselves, they can start saying that anyone calling them out is against free speech, and use that to start shutting down dessenting voices. They don’t actually care about discussing their views, most of the time they know that their views are extreme, that’s why they couch them in nicer, less honest terms all the time. They don’t care. They just want to gain as much influence and power as possible, and they’ll use any platform they can get their hands on to do it. If you aren’t aware of their tactics and how they work, then it can be difficult to spot what they’re doing, and most people haven’t done research on the subject and thus won’t recognize the dog whistles and manipulation. And their targets aren’t always the Jews, sometimes they’re other groups, like trans people, or left wing groups, or the gays, etc. They might even pick and choose based on whatever gets them more traction at the time. Jews are just their most well known and popular target. But the nazis ultimate goal as a movement/idiology isn’t specifically to get rid of Jews, it’s to gain power. The Jews were just an easy target at the height of their power.
Some examples of dog whistles that have been particularly relevant in recent years include calling an entire demographic “groomers” or “pedophiles” and associating then with pedophilia. Claiming that they’re just concerned about protecting a specific vulnerable demographic (this demographic may or may not actually be vulnerable) from a perceived threat. Their favourite demographic is commonly women, children, or in the case of more racist actors, they may be trying to protect their heritage, their genes, their population, their jobs, etc. People who tout the great replacement theory are heavily intertwined with nazi ideology for example, if they’re not just outright nazis. The target of their ire doesn’t really matter, they just need some sort of enemy or boogieman to unite people against, including people who might not otherwise cooperate or associate with them. See how nazis have been cozying up with terf groups, or religious groups, or anti-abortion groups for example.
They especially like to take advantage of economic or political strife to get their feet in the door. That strife creates desperation, makes people more susceptible to propaganda that promises a solution and gives them an easy enemy to fight against and a strong group to follow and team up with. Then that group promises that they’re the ones who see the dangers of said enemy, and will thus be the ones to actually do something about it.
Hitler didn’t get popular in a day. He had to build his credibility over time, and giving someone a platform makes them appear more credible no matter how ridiculous you think their claims might seem. And if they catch the ears of enough important people, then they can really utilize their influence.
They aren’t coming to these places of free speech to debate in good faith, they’re coming to find targets, victims, and likeminded people who will group together with them. They’re coming for visibility, and to have more ears to spread their lies to. And those lies can be very convincing to people who are in an easily influenced state of mind, like after a crisis, or an upheaval in their way of life, or if they feel like they’re threatened in some way. They’re master manipulators, gaslighters, and abusers. Maybe not every individual member, but the movement as a whole isn’t the obvious guy carrying a nazi flag and yelling “death to all Jews” in front of government buildings on the news. More often than not those guys are used as a smoke screen, so they can point at those guys and say “see? I’m nothing like him, I’m much more reasonable. So obviously, I can’t be a nazi.” even though they absolutely agree with those guys and feel the same way privately.
Providing a platform for free speech and debate only works if everyone who comes to that platform also believes in free speech, and is acting in good faith. Nazis, as a rule, do not act in good faith. Their goal is to commandeer your platform to gain power and erode the rights of others until they are the ones standing at the top and controlling what is allowed to be said. Any rare case of someone who aligns with nazi views but actually wants to debate and believes in free speech is, in the eyes of nazis, a recruit waiting to happen. Letting those people expose themselves to more organized nazis is basically throwing them to the wolves and asking for them to be radicalized.
There’s more, a lot more, too much for me to really go over in a single comment, so I recommend looking into the methods that the nazis have used both in the past as well as recent years. You will start seeing a concerning pattern in their behaviour and methods, as well as in the typical outcome if you let them weasel their way onto your platform.