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.