I don't think you're understanding what I'm saying.

Not everything leads down a slippery slope.

It can, and has. In the USA at least, we need so many ways to protect the federal and state government from interfering. We think the 1st Amendment is one of the strongest. Hell, the SCOTUS has said that hate speech isn't allowed under the 1A, but I can't think of anyone who's been arrested for hate speech alone, only verbal and physical assault while using hateful words.

The problem is that if the government has the authority to punish hate speech, you don't get to decide what is considered to be hate speech. Speech based on biological factors could be a definition of hate speech, but there are others. Would you want to live in a society where, for example, you could go to prison for criticizing religion? or a political view? You will probably say "but only the things that I think are hate speech will be punished," but frankly that's not a call that you will be in any position to make. And if you go "well no, that's protected", you're basically going around your banning to please your views, and not others.

Who decides what a 'slur' is, or what constitutes a so-protected group? Do the protections of these groups, or the sanctions against the terminology so associated, ever expire?

Do you think that a Republican and a Democrat (or even two of the same party from different regions) would agree on what groups should be protected, what speech should be limited, and what punishments should be enforced?

What constitutes infraction? What is the standard of proof? What constitutes public speech in this context? How do you consider context?

I agree that the use of such speech can be hurtful, and should be discouraged. However, I disagree that the use of this speech should be banned by law.