I don’t understand why freely speaking my mind is necessarily a “democratic right”,

Long story short: You can’t effect change without organizing, and you can’t organize without talking about organizing. And, well, a system where common people can’t effect change can’t be called a democracy. For liberal democracy specifically, how are you going to campaign and run for office if you can’t even talk about your platform?

Couldn’t an authoritarian leader simply not care and move ahead with his plans if he has enough support?

I’m not sure I understand the question.