'It is really difficult to use atDNA to prove a specific connection where shared ancestors are 6th great grandparents.'
If you have a specific DNA overlap, especially if it's greater in size than a certain threshold (7cM), it's considered valid evidence of a connection. What you CANNOT say is that NOT having a DNA overlap greater than 7cM means there is NO connection.
In other words, you cannot use atDNA to prove "no relationship" unless the generational distance is fifth cousin or less. If you have a DNA overlap, though, it is meaningful.
I keep an extensive genome map which includes DNA overlaps for both my mother's side genome and my father's side genome. From that map I can determine, for a specific DNA overlap, what common ancestors that DNA likely came from (but not the precise distance). I can also provide links to DNA relatives that are in common for that particular stretch of DNA. I've been working on this since 2019 and analyzed thousands of DNA matches and relationships, so it's pretty advanced at this point.