This is a rather frail straw man. The libertarian view that interpersonal relationships are properly based on voluntery agreement is not necessarily joined to "cash payment" (most human relationships are obviously based on other rewards).
As for it being based on "self-interest", this is true in the sense in which every political or social principle is based on self-interest (libertarians think people would be better off if they followed libertarian recommendations, conservatives think people would be better off if they followed conservative recommendations, marxists think people would be better off if they followed marxist recommendations, etc). Thus, Kirk's statement is technically correct, but trivially so and useless in weighing the merits and demerits of libertarianism.
Again, my other readings show that Kirk knows better than this, and I simply don't see what caused his usual level of argument to sink to that normally found at a bar or (present company excepted) on some Internet argument forum.
I would like to slightly modify the "every" claim. Some political philosophies are in fact based on cynical self-interest (e.g. Nazis don't claim that everyone would be better off if they subscribed to Naziism -- they believe that the Aryans would be better off, the untermenschen would be worse off, and that's simply the way it should be).
This caveat does not really change the underlying point, since libertarianism does not claim that some people are and deserve to be more equal than others.