Free speech absolutism was no defense against fascism - the Nazi Party was prominent in both the US and UK, to use your examples, during the 1930s. It just never quite tipped into power, but you can see from the rise of Trumpism that that didn't hold true this time.
https://www.npr.org/sections/codeswitch/2019/02/20/695941323...
https://time.com/4516276/cable-street-battle-london-east-end...
(the giant murals of George Washington and the talk about "job-stealing refugees" is particularly apropos given the current atmosphere of the Trumpist faction)
It is obviously correct that once social norms have reached the breaking point that hate speech laws can become a cudgel used to further hate speech. The goal is to keep it from reaching that point.
It is in fact probably correct to say that in either approach once the social norms have reached the breaking point, that either positive or negative freedoms can be weaponized. Freedom of speech can be used to rally a lynch mob, and hate speech laws can be used to oppress a minority, once you have reached the point where a hateful party has widespread social acceptance and control of a sufficient number of the political levers of power.
In general though it is also not correct to say that the US has complete free-speech absolutism. Actually planning to lynch someone, for example, is not protected free speech. The argument for further restrictions is that those lines may not be correctly drawn by our forefathers - why should actually planning to lynch Tom Brown be illegal, but the KKK rallying to lynch somebody be legal and protected, as long as they don't name a specific name in advance? Why should arguing that people should be lynched in general without the specific intention to do so today be protected? There are many places we could concievably draw the line without really impacting anyone's actual freedoms to do anything other than incite hatred and ultimately inciting violence and death.