Hmm.

You pose an interesting question and I might not have sufficient information to give you a full picture, but I might try to show a glimpse of what I think may be happening.

Someone somewhere decided it may be a good idea to expand existing framework ( hate crimes[1] ), which was relatively easily adopted in America due to historical ( slavery ) and political factors ( combating racism ).

[1] https://www.justice.gov/hatecrimes/laws-and-policies

With that in mind, the first step is saturating the media with opinions indicating some sort of support for a policy/law change ( in this case hate speech - links with sample articles follow - note how old some of those are ):

https://www.washingtonpost.com/opinions/2019/10/29/why-ameri...

https://newint.org/sections/argument/2012/12/01/is-hate-spee...

https://www.ihrb.org/news-events/press-centre/the-challenge-...

https://theconversation.com/the-idolization-of-free-speech-i...

Once the population is sufficiently primed, one can run it through congress. It is not a weekend project.

Note that it is exactly the same pattern with encryption battles, but at least that one has clear originators ( usually three letter agencies ).

Whatever efforts are happening here will be shut down by the courts unless they managed to do something about the first amendment. I am not saying these efforts are good or bad, just that they are (I guess in my opinion) extremely unlikely to succeed.

That would come up against the interpretation of the First Amendment espoused by the Supreme Court in 1969, in the case Brandenburg v. Ohio. Before then, the First Amendment was not interpreted nearly so broadly, and speech regulations at the federal and state levels were common, beginning with the Alien and Sedition Acts soon after the Constitution was ratified.

If/when the Democrats restructure the Supreme Court to have a liberal majority, overturning Brandenburg with a single ruling becomes on the table, and the door is open for hate speech legislation.

First part I agree with. The second part, maybe I guess? But that is, if ever, decades away so we can worry about that when it becomes a realistic prospect.

I don't think any of the current sitting life time appointed supreme court justices have expressed any will at overturning that decision.