Your message certainly caught my attention!It is pretty well concluded that Kit Carson was the son of Lindsey and that Lindsey Carson was born 1754 (eldest son of William).Lindsey was quite old to be a father (his youngest Lindsey Jr. born in 1818 when Lindsey Sr. was 64) but his second wife Rebecca Robinson was much younger born ca 1778.
In Kit's autobiography he lists his POB as Madison Co., KY (confirmed by 1860 NM census) The only Carson appearing on the 1810 Census for Madison County was Lindsey.Several siblings and half siblings of Kit claimed him to be their brother.Most helpful were the comments of Kit's half brother Moses Bradley Carson (b 1792).He served with his father Lindsey in the war of 1812 and claimed that Kit was his half brother. Since Moses was born in 1792 and Lindsey's eldest child William was born 1786, Lindsey could have been born as late as 1770.However Lindsey was noted as the "eldest son and heir" of William Carson in documents dated 1784.It is doubful Lindsey could have inherited if he had been younger than 18 so he had to have been born no later than 1766.Not to mention his younger brother Robert is reported to have been born 1758/59 (married Lindsey's sister-in-law Jane Robinson)
I have recently come into a letter from Christopher Houston to Lindsey's brother Andrew Carson, which informs him of his brother's death from a falling tree branch.Christopher Houston being the namesake for Kit.
There might be more proof out there but I don't have it at my fingertips.Certainly several prominent researchers have concluded Lindsey to have been Kit's father.I hope this helps.I'd love to know your thought and interest in the Carson family. Doug firstname.lastname@example.org