In my understanding, this is the main reason for separating "hate crime" from all other violence
There's nothing wrong with determining the motivation for any crime, but if that's the reasoning for penalty enhancements then it's a poor reasoning unworthy of creating an imbalanced system of punishment and inequality in who deserves what justice. The prospect of incarceration doesn't decrease criminal activity, and stricter penalties don't either. In the US, adulthood brings along much higher odds of prison sentences and stricter penalties, though studies have shown there's no decrease in arrest rates connected to turning eighteen.
It's not so much about determining the reason - it kindov is more about additional punishment:
You are punished for violence-against-anybody
and
You are also punished for incitement for further violence - by having visually, clearly and deliberately chosen your victim from a political/ideological point of view, and knowingly made so publicly.
Two-in-one. I agree with how you think, don't misunderstand, I've thought the same many times, crime is crime, violence is violence, a victim is a victim.
Think of free speech as a similar thing - you are not only expressing your free opinion, but you are inspiring others to do crime. What if my hate speech is very direct? "Go buy a rifle today! Start hunting brownies!
Cut the tall trees! Cut the tall trees NOW!"
That was the code-word to commence the massacre in Rwanda, for example - was that merely free speech? Of course not, it was a pre planned terrorist event of unimaginable proportion. Can you imagine free speech unleashed in Germany, where they
explicitly ban all and any Nazi propaganda? I can assure you, it would look different than a KKK rally, these guys actually had the practice, history and memory of a Europe engulfed in flames, and they want it back.
Then it is needless to add - I obviously agree with a general free speech, but surely you can see how at least some places may prefer to limit it, for the sake of national security - people aren't always in control, speech may incite them.
And, if a "hate crime" is meant as an example to go by, as they sometimes are, many of the perpetrators admit to wanting to ignite a race war or something, they should be treated as more than solely the act of violence.
Maybe it's just a bit poorly defined, as a law. Maybe it could be just violence + incitement to terrorism, for example, instead of "hate crime"