says that the most efficient and non-market distorting way to get the users to pay the cost of the externalities is to impose a carbon tax.
Hmm... please consider this statement carefully.
Lets assume that these two things are true:
(A) Carbon emissions has an "external cost" associated with it, and that this cost is born by the people of earth.
(B) A Carbon tax can be levied that is approximately equal to the "external cost" of (A)
Your statement is still not correct. Paying a tax to the government is not at all the same as paying the external cost born by people, even if the tax is exactly equal the cost.
Your argument is essentially the same as if I accidentally burned down my neighbors house, that instead of buying him a new house I have to hand over an equivalent amount of money to the government and then he gets whatever the government decides that he will get.
Do you feel that if I do pay the government an amount equal to a new home for my neighbor that I have paid my neighbors costs? Really? yeah, I didnt think so.
The problem here is that "external cost" is a nebulous thing which allows you to be equally nebulous with your thinking about what "paying external costs" actually means.
Carbon taxes do not pay the external costs of carbon emissions. Full stop.