Hate speech is a nebulous term, extremely dependent on the speaker's values.
Plus, people aren't really obliged to love one another or their institutions. Why should I pretend, for example, that I respect some dead Iron Age prophet and people who follow him like sheep?
And yet anti-religious speech is usually perceived as hate speech by the religious folks on the receiving end.
'Hate speech' is a term deliberately coined to undermine the ideal of Free Speech.
There is no need for the label, other than to serve this purpose. We already have long existing words for what is lumped under so-called 'hate speech', such as bigotry, or invective or slander. But they don't contrast so neatly against Free Speech as the invented (subjective) label 'Hate Speech' does, which is why it was (only recently) coined.
My point on spectrum wasn't relating to free speech but rather free speech regulation.
Open discourse is universally recognized as a generally good thing for society.
So, we can try to embrace Free Speech ideals even as we grant our government right to censor some speech and draw (a somewhat arbitrary) line of what's allowed.
We've seen Russia, for example, abuse that power even though free speech is written into their constitution, they use that power to censor political speech and what they refer to as blasphemy, so we know this power can and has been abused but it's also possible, so, perhaps, you are right.
It also happens to be an ideal that ppl will vigorously promote in certain domains, while betraying / discarding it in others.