Does the government of USA grant rights or simply recognize rights that are always there?
The rhetoric is irrelevant. It does grant rights. The slaves sure did not have those "inalienable rights" that were "endowed by their creator", they had no rights until the US government GAVE THEM rights. These rights were also oddly missing in many parts of the country under Jim Crow laws, until the Federal Government gave them their rights. Rights for women were surprisingly absent for many centuries. The right to marry the person of your choice is still under debate.
So no, I don’t buy the rhetoric that the rights are already there, we’re just "recognizing" them. Yes Government does give rights.