"Aaron and Kathy are 4th cousins six times removed" We are connected through John Warham, who was the founding Puritan preacher in Windsor, Connecticut in the 1630s. Warham is my 9th great grandfather Aaron Burr's 3rd great grandfather through the famous Great Awakening preacher Jonathan Edwards.
My hunch is that it's extremely likely that a Burr researcher could find much closer links by connecting existing profiles to descendants, since both of Aaron Burr Jrs were early migrants to Connecticut, linked to prominent families in the Connecticut Valley.
But this might also be a great opportunity for to explore two other likely options of closer connections to WikiTree members if a genealogist did a bit of online research:
A. Research possible descendants of Aaron's sister Sarah Burr Reeve (see comment on Burr-111 profile).
B. Document updated research on Aaron Burr's two children with Eugénie Bearhani. Each had children who survived with adulthood: Louisa Charlotte Burr (Burr-948) and John Pierre Burr (Burr-949).
Looks like the Burr Society has checked the family trees and accepted these two children and their descendants based on the evidence provided, and there are even books that came out on this in connection with the 400th anniversary of 1619 (https://www.theguardian.com/us-news/2019/aug/24/aaron-burr-vice-president-who-killed-hamilton-had-children-of-color). What a positive step it would be to incorporate this documented connection on WikiTree!