Considering how hate speech is a very flimsy term, I don't think that this always follows. I do believe that there's likely a correlation, but I doubt it's strong. Some will say that claiming that there are only 2 genders is hate speech, others that claiming that abortion is murder is hate speech. Not to mention that there are many cases where something clearly not hate speech is labeled as hate speech.
In the UK a teenager was arrested for quoting rap song lyrics on an Instagram profile page. This was labeled a hate crime by the prosecutor.[0] Do you think that this makes her more likely to engage in hateful behavior?
Bullies at school often get away with their bullying, because they bully in ways that are not overtly noticeable. They will use terms that aren't offensive to make fun of and hurt others.
Everyone thinks hateful thoughts in brief moments of anger, but very few people ever act on those thoughts or express them. Their whole recruitment process was to get someone to move from just thinking thoughts to acting on them, usually by spreading the message to others. Once you cross that line from thinking to action, the bright line divider is gone--it's all just a matter of degrees of action.
Making it harder to cross that first line absolutely matters in making it harder for them to cross the line of becoming an active danger to others.