Couple of facts:
1) De-platforming, as far as anyone has looked into it, seems to work. There might be a short Streisand-effect, but long-term it seems to be quite effective. (Obviously more research is needed.)
https://techcrunch.com/2017/09/11/st.../?guccounter=1
https://www.vice.com/en_us/article/b...edia-bans-work
2) When it comes to inciting violence, literally inciting violence isn't the only or even necessarily the most dangerous thing one can say. De-humanizing and othering is. Once you sufficiently de-humanize a group or label them as "the other", they will pretty much automatically become a target of oppressive or bigoted action whether anyone suggests it or not. (This type of speech is btw the type everyone should be watch out for in their own language. Few will say Trumpists should be shot, many will on a bad day call them scum. It's not the same, but it's still dangerous.)
There is no one link for this, but the research relating to dehumanizing speech is rather extensive and its connection to increase of violence and political oppression is quite clear, although obviously not direct.
Therefore supporting freedom of speech for dehumaning speech is wrong, unless you are indifferent to the actual violence and oppression it causes. There is unfortunately very little middle ground there.
The idea that hatespeech regulation is somehow indicative of the society becoming more puritan or overly sensitive morally is ignorant. It's the equivalent of saying that strictly limiting the use of asbestos happened because of oversensitive environmentalism. Neither is true. When we learned just how dangerous asbestos isz we started to limit its use.
Societys have started to put limits to dehumanizing talk because we have discovered through research that it's actually very dangerous to people's health.
Edit :
Much of the groundbreaking in our current understanding of hatespeech happened while people studied the genocide of Rwandan tutsis in 1994 and the Yugoslavian Civil war, both situations where previously seemingly peaceful coexistence erupted into horrific violence and persecution. It's been a long process to turn that understanding into legislation, and it's likely not perfected yet, but in general I would consider this a case where science is winning over ignorance.
We didn't previously really understand just how dangerous hatespeech is, but now we do, and we're starting to do something about it.