For natural science, to prove the existence of something means to manifest it to sensation. Regardless of whether one is trying to prove the existence of black swans or black holes, the proof is basically the same: you introduce something into sensation that wasn’t there before. You can point to a black swan for the first time, or point to some not yet seen effect of a black hole in a picture taken by a telescope, and viola, you have proved the existence of something.
In proofs for God’s existence, nothing new is introduced into sensation. The universe viewed as a “thing” and a universe viewed as as a “creature” yield identical pictures, experimental results, relations between events, etc. If it is obvious that “to prove the existence of X” means “to manifest X to sensation” then it is obvious that there are no proofs for the existence of God. When I hear people speak of it being simply obvious that there is no God, and that there is simply no evidence for him, I suppose they are thinking something like this. For what it’s worth, no one who claimed to prove the existence of God claims to have that kind of evidence. Along with St. Augustine, everyone insists that nothing giving that kind of evidence is the divine nature. If you can imagine something, it is not God- at least not as the theistic proofs speak of the divine nature.
Proofs for God’s existence are based on sensation, but not in such a way that the difference between proving something and not proving it means being able to catch it on film or not- the way we can film a black swan or the trace on a metal plate that proved there were sub-atomic particles (there was a good deal of theory involved in showing that a sub-atomic particle would leave “trace X on a metal plate”, but the moment of truth is the trace on the plate, not the theory).
What the theistic proofs have in in common with the other kinds of proofs for existence is they are all necessary because of some weakness of our intellect. No one needs to prove the existence of trees since they’re just there. For the same reason, no one needed to prove the existence of black swans to Australians; and no one would need to prove the existence of black holes to a civilization which (somehow) could just look up and see one in the sky (like the passengers on the Disney movie The Black Hole). The difference is that the standards of what counts as proof are different in the case of natural science and in metaphysics.