Thank goodness we are still able to make the case. The only alternative to free speech is violence. I want to be able to hear what the alt-right has to say, what Islamic extremists have to say, what misogynists have to say, etc.
I want to be able to hear what all those people have to say, so I can try to persuade them that they're wrong.
Yes, persuade people they are wrong where they are wrong; learn from them where they are right (we are all fallible). Persuade other people that they are wrong; teach other people where they are right. Plus an unlimited number of other possible responses.
>I can try to persuade them that they're wrong
That assumes you know what their ideas are. If present trends against free speech continue, it's not a luxury you're necessarily going to have.
Emotional responses is not logical discourse. Also, morals, norms and values are very hard to change. I mean most people - including you I believe? have a moral saying "Murder is wrong". I don't know how to convince you to change that moral.
But, when it comes to issues inciting a strong emotional response, that becomes easier. "It's wrong to punch people in the face" becomes a much more flexible norm when said person is a nazi. "Murder and the death penalty are wrong" is a norm that becomes looser when it concerns someone that raped your child.
TL;DR, persuading someone off their values and moral codes is not trivial. Those are things that take a long time, gradual and long-term campaigning, subtle cues of acceptance. They do work, mind you; 70 years ago, homosexuality was a crime and the punishment was castration (Turing), nowadays it's much more accepted and gradually becoming normal.
I think this is really difficult, but a worthwhile goal. In all honesty my goal would be to understand where they are coming from and help turn them or future generations from such course.