Regarding the main topic, I would agree that religious tests for public service are not constitutional. I agree in principle with several points in the video, even though I am no longer an atheist. I did find the fellow painting with an awfully wide brush of "religion" as his vague antagonist, a tactic equal to those used in any traditional religious rivalry.
Regarding the peripheral discussion of theism vis a vis atheism, it is hard to imagine that conclusive verification will ever be found from nature to "prove" the existence of the Creator.
On the other hand, there is a lot of "evidence" that points to the possibility. What perfect luck that the moon, only 240,000 miles away from earth, appears the identical size in the sky as the massive sun, 93,000,000 miles away. The sun is 388 times farther away from earth, but the moon's disk perfectly covers the sun during solar eclipses. Not only that, but the orbits of the earth and moon permit eclipse shadows to fall on earth, to be viewed by sentient beings that can appreciate and marvel at such a lucky chance. Well, well. Lucky, lucky us. No other planet-moon system in the solar system attains such a perfect eclipse.
Proof vs possibility. I think that is a key difference in view here. Proof? Not conclusive. Possibility? Fairly good.
As a young atheist I wanted to know two main things.
What is the point of existence?
What is the point of my life in this existence?
During the time that I was pondering these questions, I was learning in Calculus that any finite number compared to infinity appears so small that it approaches zero in comparison. If my transient finite life of 70-80 years was going to wind up in everlasting annihilation, then what really is the point of it all? That question led me to the "possibility" that there may be an eternal Creator, and following from that, an eternal aspect to life. Not proof, but possibility. From then on for several years I went through no small struggle to find more clarity about these issues.
So the bottom line for me is that I found a personal, intimate relationship with the Creator via channels in my life that I find more real than this bulletin board. My relationship is not based on what someone else tried to prove to me, but based on my personal willingness to communicate with the Creator as a submissive creature. I certainly cannot prove to you that God exists. Au contraire! You would have to let Him prove to you directly that He exists, as a submissive creature.
And if you are not interested, hey, free country... Just to add that for a person of sound mind who persistently ignores the opportunity to meet the Author of life, there may be a persistent consequence unrelated to the constitution.