You've touched on something I had considered addressing but thought might end up with posts that are too long.
It all comes down, in one respect, to the fact that we all must believe something is eternal. You chose "existence", a deist would say "God". God makes more sense when looked at logically. To believe in God as the "eternal" takes less faith. I will explain...
A belief in existence as eternal requires many assumptions. Firstly, it must be a 100% closed system, if even a fraction of energy/matter is lost then over eternity the whole will wind down, thus making itself non-eternal. (this would be in keeping with the laws of dynamics, but science, to maintain their view must argue that it does not apply to existence) Scientists posit an expanding and contracting universe with periodic explosions as the mechanism for each expansion. What was the cause for the very first explosion? Everything tends to settle to a point of equlibrium and rest unless there is an outside influence.
There is no proof or even evidence that the universe is a 100% closed system, or does science have anything approaching an explanation why the universe, given infinite time would continue to be in motion, these views must be taken on faith. Anther set of dogmas of atheism.
If one believes that there is an eternal God, who is outside of the physical laws (being the creator of those laws), then there is explanation for the appearance of the "wind down" nature of the universe, as well as the prime mover of the universe. This logically makes more sense than the best science can offer, thus requiring less faith.