That argument is equally applicable to both sides. There are some right wing people who foam at the mouth or commit violence at the idea of gay people marrying. On the other side, you get some left wing people who cry and riot and call for male genocide because of things like manspreading or being misgendered.
Violent conduct and suppression of free speech isn't acceptable from either 'side', regardless of their motivations.
Well yeah, but it's not like there are legitimate left-wing politicians calling for manspreading to be a criminal offence. Can't say the same for right-wing politicians and being gay, for example. That's the difference between individual and systemic prejudices.
As an aside, I do dislike when people claim suppression of freedom of speech from private citizens.
That's not a thing. Only a government can suppress freedom of speech, because freedom of speech is freedom from government oppression. A private citizen is completely in their right to deny another person freedom of speech in private situations. Private citizens have
zero obligation to provide a platform to people they disagree with. There are arguments for doing that, but it's
not a fundamental right.