This is all semantics, of course, but I'm not suggesting that nothing can be proven. I'm saying that DNA is simply another piece of evidence, that needs to be weighed along with other evidence. I've debated a couple relationships that people claimed are "proven" because of a DNA match.
Yes, if the mailman has no siblings and it was not possible for his father to be the parent of either your 3x or 4x great-grandfather then the mailman it is. However, that is using corroborating evidence and argument.
And you'll have to show me on the Big Tree where there is a tree in the genealogical timeframe, where the DNA alone proves an ancestor. I use a Y-DNA test to prove my 8x great-grandfather myself, certainly. But I also use a land indenture where he refers to my 7x great-grandfather as his son. Otherwise, it would be hard to know for sure. Did Jefferson father those children or did his nephew? They needed to make an argument for Jefferson based on available records.
I also use an autosomal DNA network as evidence in a proof for a 5x great-grandfather and a network for my wife's 7x great-grandparents.
I think in all these cases, DNA is an important or even critical part of the proof. Here, though, I can only use DNA confirmed for the Y-DNA because they decided DNA networks aren't "confirmation".
Again, I'm simply questioning the value of saying that DNA proves anything. DNA is a single piece of evidence.