There are connections between the two though, annoyingly often. Advocating for a policy change that will ruin some people's lives. Advocating for a status quo that will ruin some people's lives. Advocating for some group of people to be fired. It doesn't make sense to see speech just as a consequence-less thing, in both directions.
You have to draw the line somewhere, right? Otherwise you get into slippery slope territory where eg. you can argue anyone who's not protesting for BLM are implicitly supporting the cops, and therefore are ruining black people's lives.
Sure. Not everything warrants a reaction, impact of speech highly depends on the speaker and context and reactions can be not proportionate. But I don't think a blanket "speech shouldn't have consequences" really works, because it will often have them, and often is intended to have them.
But speech and opinion-forming plays naturally a large role in it, especially in a democratic system. Lots of speech does have consequences (and lots of it is intended to have consequences that aren't just speech without further consequences). Speech shouldn't have consequences just doesn't work as an absolute position IMHO, and if it isn't then it's harder to explain that counter-speech isn't allowed to have consequences. (I'm explicitly not saying that every consequence is acceptable or justified)