In the US you must opt-in to a religion. Parents can raise you according to their religion,the bill of rights applies to government not parents so they can tell you whato to say do and believe. Basic human rights are protected for most people (except prisoners I guess),but also additional rights for children exist both at the state and federal level. The idea of being born into a religion is not recognized. The declaration of independence is apparently not a legal document you can use in court. If it was,the fundamental rights you mentioned according to that document are self-evident and are given to mankind by God, meaning government cannot implement laws against the self-evident will of God and the self-evident rights of God has given to mankind. It was the foundation of their argument for independence, since after all King George is legitimate only because the people believe he was divinely appointed,deriving authority from God.
This idea that democracies exist in acknowledgement of God and derive legitimacy not simply because they represent the people's will but because they protect the people's God-given rights(even for the minority) was well accepted well into the past century. Some even reallized how the atheistic views of communism is incompatible with democracy.
Even moving past independence, could Abraham Lincoln have made the argument that slavery must be abolished due to "all people are created equal" if he was a communist ? That argument was relevant because it was once again the government taking away people's God-given rights.
If you wonder why your rights are treated so cheaply and all the freedom you have is fading so rapidly is because no one,not the government,not the majority of people or the few that fight for rights acknowledge that the foundation of these fundamental rights is God's authority, that a government and a country while sovreign relative to other countries,there is only absolute sovreign from whom all authority and rights are derived. But going back to what you said, the fundamental rights might as well be fiction to the legal system and SCOTUS,the constitution itself is the document they think is "a living document" they can interpret as they see fit without regard to where absolute rights and liberties originate.