Being a descendant of a long line of blockaders from the mountains of North Georgia, I know a little bit about how whiskey was produced under primitive conditions. While it's true that there was a variety of spirits around in colonial times, most of the beverage alcohol in a particular location was made from what was available at that location. Transporation was expensive, and people on the frontier had little money to import commodities that they did not grow or make at home. So, they made whiskey from what they had.
Because molasses from sugar cane was available on the coast or in the very warmest areas, they made rum there. On the frontier in the east, the only materials available to make whiskey in quantity were likely corn, wheat, and maybe some rye, barley, and sorghum. So, this was likely what was used.
I think that corn whiskey was likely the most widespread simply becasue corn was the primary crop of most frontier farmers. Both bourbon and Tennessee whiskey started out as varieties of corn whiskey and corn is still the primary ingredient of both. The bourbon that we know and love is a variety of aged corn whiskey, but on the frontier, aging was an unecessary frill, and they likely drank it as it came from the still, or maybe cut it with a little water.