The emphasis is heavy on democracy because that's the basis of our society. Rule by the majority. Rights are narrow exceptions to that general rule. That's why the bulk of the constitution talks about implementing the democracy, not rights.
The 9th Amendment (which is essentially redundant in that it would be true even if it didn't exist), states that 'The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people.'
The point is, Rights are not exceptions. It is the removal of a right that is the exception. In other words, you have the right to do anything unless it is made illegal, not the other way around.
Moreover, while it's true you can do what you want unless it's made illegal, the interesting question is instead: what can the majority make illegal? That's is the situation where majority consensus is the rule, and rights are the exception. In a democracy, the majority can legislate however it wants, limited only by certain acknowledged individual rights.
In our democracy, we have two layers of legislatures, state and federal. The state legislatures inherited the sovereignty of the English King, and the constitution did not disturb that arrangement except in that the states delegated certain of that sovereignty to the federal government. But at the time of the founding, the power of the state legislatures was almost unbounded. They were not even limited by the rights in the Bill of Rights, which weren't applied to the states consistently until the 20th century (incorporated via the civil war amendments). Even now, the state governments are emphatically not ones of enumerated powers. The majority can do whatever it wants, limited only by federal law, or state or federal rights.
At the federal level, within the broad enumerated powers of the government, the majority can do what it wants, limited only by federal rights. Again, the rule is majoritarian control. Rights are a limited backstop to that general rule.
If anything, American education focuses too much on rights and not enough on democracy. K-12 curriculums tend not to cover any state constitutional law. They focus on enumerated powers and the Bill of Rights. But in doing so they paint a misleading picture of the Constitutional scheme. A scheme in which, as originally conceived, nothing would've prevented states from making it illegal to have sex with the lights on or outside of marriage or to use birth control or any of the other "moral" legislation that was very widespread at the time of the founding. I don't think expansion of rights of individuals has been a bad trend away from the original constitutional scheme, but it's misleading to paint the trend as pointing in the other direction. That characterization is simply a libertarian rewriting of history.