I have just started doing some reasearch on The Burtis family on my side of the family.Here is some information that we have that may or may not help you.My wife's line comes from the Jesse Burtis line that you referred to.The information we have is as follows:
Jesse Burtis, b. 12/12/1780 in Long Island NY.Married Elizabeth Miller (date unk.) d. 12 Sept. 1853 in Vanderburgh Co., IN.
Children: 17 Sept. 1805 Daughter Susan Moser, NY 2 Nov. 1807 Son Edward, NY Sometime 1809Son John.This could be your John.Your birth date seems incorrect as he would have been only 12 at date of marriage. 1 Feb. 1815 SonStephenMy wife's gg grandfather. 2 Mar 1822 Daughter Ann Charlotte B, IN
I hope this is of some help.Also, I have not been able to go any further back than Jesse Burtis on this tree.He was probably connected to the Alberti family of NY but I have not yet found the connection.Good Luck