If you reject the idea that people are intelligent enough to think about politics for themselves then you reject the idea that they're intelligent enough to govern themselves. Democracy and free speech are intrinsically linked.
Why is it that we have to ignore nuances in these arguments? We never had unrestricted free speech in this country. There is a whole Wikipedia article about it: https://en.m.wikipedia.org/wiki/United_States_free_speech_ex... In addition other democratic states have strict limits on free speech such as Singapore: https://en.m.wikipedia.org/wiki/Article_14_of_the_Constituti... After all even “democracy” is nuanced here…
One could also argue that censoring one side of politics deepens divisions and drives people to polical extremes. I would rephrase "democracy can’t survive without free speech" as "democracy can’t survive without free political speech"