I agree with your points. I just want to add that what OP was talking about is that the existence of a deity or higher power is not falsifiable and thus is impossible to logically prove. I’m sure many, many, many people have tried on both sides.

My favorite proof against any higher power is from the Hitchhikers Guide to the Galaxy:

Now, it is such a bizarrely improbable coincidence that anything so mind-bogglingly useful could have evolved purely by chance that some have chosen to see it as the final proof of the NON-existence of God. The argument goes something like this:

“I refuse to prove that I exist,” says God, “for proof denies faith, and without faith I am nothing.”

“But,” says Man, “the Babel fish is a dead giveaway, isn’t it? It could not have evolved by chance. It proves that You exist, and so therefore, by Your own arguments, You don’t. QED.”

“Oh dear,” says God, “I hadn’t thought of that,” and promptly vanishes in a puff of logic.

“Oh, that was easy,” says Man, and for an encore goes on to prove that black is white and gets himself killed on the next zebra crossing.