Even the idea of “banning free speech” that you mention is implemented in a liberal fashion in the US. There are rarely calls for the government to actually ban speech via laws. The ground where that’s fought is actually “should private companies broadcast/highlight via algorithm the speech of individuals who say things I don’t like,” it is a formulation that pits the speech rights of the corporation against the self-expression of the individuals using their services.
The framing you quoted (“these liberals who want to ban free speech”) is often used by one side to pitch the other side as falling outside the traditional free-speech consensus.