"If we just changed the rules and the public didn't change, imagine what would happen. Also, note that people with political causes invest enormous resources in manipulating public opinion."
You cannot in fact change the rules if there hasn't been a shift in the public so this seems a rather moot point. I'm not arguing that you shouldn't invest in PR and education on these issues but muzzling the opposition just gives the next rulers in charge an already existing muzzle to use back on yourself.
There is a history of often murderous suppression of "radical" groups in the US and by and far the majority of these were socialists, environmentalists, and marginalised people.
The current fake appeasement to these demographics is just that - a mirage. They're willing to ban Trump from the internet but not raise the minimum wage, abolish the drug war, really come after companies for their industrial scale damaging of ecology or cease trade relations with countries like China, Saudi Arabia, and Israel.
Excuse me for not taking this current "wokeness" fetishization by fundamentally hierarchical power structures entirely as good faith.
It's also not clear to me at all that "speech plays the largest role". Of course, existing power structures want you to believe that because the thing they fear the most of all is armed and genuinely effective resistance and it also helps perpetuate the illusion of a free and democratic society as if all it took to conquer racism and imperialism was to have a sit down and chat - with the hidden assumption that prior exploited people were just too stupid to consider this before - but MLK and Ghandai would have gone nowhere if there weren't decades of riots and militant struggle preceding them that pushed the US and British governments to enter negotiations with these parties.