> Most Americans have been brought up to believe in shared beliefs like individual liberty, equality before the law, basic human rights, etc.
This is less true when you get beyond platitudes to what each one of those means in substance.
There is a very significant disagreement over what “liberty”, for instance, means, and over what things are “basic human rights”, among people who agree that “liberty” and “basic human rights” are important goals.
> But beliefs are not the same as values.
Well, beliefs include both fact and value beliefs, but the ones you enumerate are, in fact, values.
> The divide comes in how we prioritize our values in achieving those goals.
Values are root goals, not means of achieving goals.
> I disagree strongly with your last point. Almost everyone wants to be happy, avoid suffering, have healthy and harmonious relationships, to live life as they choose, have a sense of purpose, etc.
Taking that as true, for the sake of argument, that's not wanting the same things, even if people might use the same words. It is a different thing for Alice to be happy, avoid suffering, have harmonious relationships, live as she chooses, and have a sense of purpose than it is for Bob to have those things; it may even—depending on, e.g., what makes Alice and Bob each happy—be mutually incompatible things.