This argument makes me pretty uneasy, since it seems like it can essentially be used to censor whatever you want. If e.g. the people fighting for climate justice get branded as ecoterrorists, wouldn't removing their 'propaganda' be ok under that line of thought?

I think the right to free speech isn't some enshrinement of the right to spew garbage, but the realization that restrictions of free speech can very easily be turned against 'good' causes.