It's commonly taught in grade and high school civics classes that, since the Declaration of Independence, the US has a tradition that certain rights are unalienable. It's a direct statement that there is no way to separate or sever those rights from a person.

It's a whopper of a run-on sentence, but it's in there: We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.--That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, --That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness.

This still doesn't make sense.

Athens itself killed Socrates for violation of speech laws and yet they are considered a democracy. This would be a violation of the First Amendment which would be considered an unalienable right that the Declaration of Independence was talking about.

There is an interesting point, "deriving their just powers from the consent of the governed". If the governed consent to an authoritarian law, why would they cease to be a democracy? If 100% of voters voted to ban wearing the color green, they wouldn't be a democracy?

I think you are basically saying democracy means good policies and authoritarianism means bad policies so you cannot have authoritarianism and democracy, but that just isn't the definition of either of those words.