The point as it relates to the American Constitution is that that it was conceived with the notion of these divine rights and explicitly recognizes that there is no authority that can deprive the individual of them, thereby placing a hard limit on what a government can do.
You're free to disagree with the notion, of course, but it's worth understanding the foundation.
One does not need to know the specific identity of God to justifiably believe that rights come from God. Suppose that I receive a handwritten letter with no name on it. By the nature of the letter, I can reasonably infer that it was sent by a human, even if I don't know what specific human it was.
GP's argument is that the nature of rights implies that they must come from God. This is because they think rights can't be taken away by others; if they could, they would be privileges, not rights. They presumably think that for a right to be inalienable, it must come from an authority above all others, like God.
You seem to think that rights only apply to specific people at specific times and places. That's fine, but it's the very point that GP was addressing—if rights are given by the government, then they're not rights at all. Restating the claim that rights are not universal does not address GP's argument.
I don't think GP's argument works when it comes to God, because it might be that rights simply exist independent of any authority. Maybe they're an emergent property of human beings, or maybe they simply exist, the way that many believe that God, the number two, or the universe itself just exist without cause. GP might not agree, but it's certainly coherent to believe in inalienable rights without believing in God.
Because the truth is - there is no "god" in the way humans think there is. Saying some mythical sky-daddy grants a certain group of people "rights" at a given point in time is laughable at best, and deliberately disingenuous at worst.