I think it's possible to make this determination with autosomal DNA, it's just statistically extremely (more like overwhelmingly) difficult.
Even with 4th cousins, there's only about a 50% chance that any two at random share any significant DNA at all. So, if you test and don't match, it doesn't necessarily mean there's no relationship. If you compare your sample against 2 actual 4th cousins, there's a 25% chance of no match, against 4 a 12.5% chance of no match, and against 8 a 1.5% chance of no match. So, you would probably need to compare against about 6 possible 4th cousins to be reasonably certain (97%) that no match indicates no relationship (this presumes that you know with some certainty that those 6 are closely related to each other).
Obviously, the numbers get a lot worse with 7th cousins. The 50% likelihood of a 4th cousin match falls to 1-3% for a 7th cousin match. That translates to optimistically on the order of 100-200 failed matches against possible 7th cousins to be reasonably certain of no relationship. The converse is also true - if you have 100-200 samples in the "known" branch of the tree, and a suspected 7th cousin matches none of them, then most likely there's no biological relationship (I'm ignoring the NPE factor here, but you'd also need a lot of samples in both lines to rule that out).
Even if you do luck out and get a match, you have the problem of pedigree collapse - a match between two individuals, one in the "known" line and one in the "target" line, doesn't provide absolute certainty that your proposed common ancestor is the source - it's possible that there's another common ancestor somewhere in the mix (often typical with ancestral groups who migrated together and/or lived in close proximity for generations, attended the same church, etc.). To rule out (or at least reduce the likelihood) of that possibility, you'd again need to have samples spread out across multiple descendant lines on both sides. I can't guess in your case, but on AncestryDNA (presumably the largest sample database in the industry), my largest "ancestor group" has about 20 members, with perhaps 2 or 3 times that many who are shared matches with one or more of them. In many of these cases, there's more than one common ancestor involved, so it's difficult to draw any conclusions about shared matches who have no listed pedigree.
A chromosome browser and the corresponding ability to map specific DNA segments to specific ancestral lines would obviously help considerably, but it doesn't change the statistics all that much.