I have been going through your line, generation by generation and I cannot find anything wrong with it.
For the earlier generations, leading up to the migration to Ohio from NJ, History of the Jackson family of Hempstead, Long Island, N.Y., Ohio and Indiana does a very good job of tracking each generations and has supporting evidence such as wills, for each step until Maj. Benjamin W. Jackson. Also, books such as Genealogical and Biographical Sketches of the New Jersey Branch of the Harris Family, in the United States expand a few more generations in Ohio, and we can support their connections with Benjamin Jackson's will, in which he names his son, Ziba. https://www.ancestry.com/imageviewer/collections/8801/images/005430...
Ziba's biography in History of Knox County, Ohio, Biographical sketches (pg. 704) attests that he a child Shalon (Chalon), and from Chalon the line is well documented to your father.
In short, as far as documentation is concerned, I see no issue with your line going back to Robert Jackson.
I also looked at Darrell Jackson’s ancestry, and his line is much sketchier. His Jacksons were from Tennessee and the problem occurs in the jump between his ancestor, David Jackson of Campbell County, TN (see https://www.wikitree.com/wiki/Jackson-30732) and William Jackson, who may have moved from Morris County, NJ to Carter Co. TN. This William Jackson of Morris Co, NJ is the son of General Joseph Jackson, your ancestor. There seems to be a bunch of research here which you can find on Wikitree https://www.wikitree.com/wiki/Jackson-22052 and https://www.wikitree.com/wiki/Space:Study_of_various_William_Jackso...
The important section: “Darrell Jackson, a descendant of this William Jackson has participated in the Jackson DNA Project at FamilyTreeDNA. His DNA is a 99.9 percent match to a well documented, known descendant of Gen. Joseph Jackson. So this is further verification that William who died in Carter County is indeed Gen. Joseph Jackson's son.”
A couple things stand out to me here. First of all, they say his DNA is a 99.9% match, but they do not really go into detail on what that means. Does it mean that a DNA service matched them as cousins, or do they share the same Y-DNA? It is not clear from this statement, and the difference is a very important one. What is clear, is that the documentation for this connection is simply not there. There are lots of references to Jacksons in NJ, NC and TN which they are trying to piece together. Unfortunately, Darrell Jackson is not longer with us, but if we want to pursue this further, I think that the next step would be to try and find which “known descendant of Gen. Joseph Jackson” did Darrell Jackson match with, and more specifically, exactly what was the nature of this match.