It's not pedantic. One of the most notable aspects of the US constitution is that its power is explicitly derived from the masses - "We the people..." - and not from some divine power, unlike the British monarchy.
Power is derived from the people, not rights. An early argument was over the very need for an explicit bill of rights. Even the text of the bill of rights suggests that the founders believed that certain rights are inherent:
Congress shall make no law...abridging the freedom of speech...
That wording seems to just assume that people have the right to speak freely, that no law is needed to give people that right, and that laws can only restrict that right (and such laws are explicitly forbidden). The phrase "God-given rights" is not meant to assert that God exists, it is meant to assert that certain rights transcend legal codes and exist regardless of what government happens to be in power (or even if there is a government in power).
