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> Plenty of people are being harassed in public for their sexuality, race, etc.

Harassment, regardless of motive, is illegal. Is there a need to make it more illegal?

> you don't need a 20 years long control study to prove that outlawing hate speech will improve their rights.

Restricting a collective right does in no way grant more rights to a minority. It may (superficially and short term) improve their lives but do you really thing people will be less hateful from being told they are hateful? I always found that conclusion odd

"Hate" is fuzzy. Some Jews believe hate is any kind of critique of Israel. "Racism" is fuzzy. Some black people believe it is racist to dance a certain way if you're not black. "Sexism" is fuzzy. Some women believe they have been violated by a compliment.

Definitions change. Most of the time irrationally. What was well-intentioned yesterday becomes hateful tomorrow.

This is a slippery slope my friend. Not many things in life are absolute but freedom to speak your mind must be one of them.