>As my ECON 101 teacher said many times, the way to handle negative externalities is to tax them. The free market cannot handle this.
To take your second part first, taxing is part of the free market. The free market is a creation of and tool for use of government. Regulation to ensure cost internalization, information symmetry, contractual enforcement, and so forth are all basics for it to work at all. And a market has no goals of its own nor does it take into account human floor levels, it's up to society (and in turn its government) to choose goal states and then use all tools to find the most efficient routes towards reaching and then maintaining them.
To go back to the first, you've just restated the question: how do you "tax" the externality? How do you find the right rate? How do you adjust it and account for changing factors? It's government that sets the goal, but government tends to be bad at figuring out implementations and then reacting with high temporal or spatial granularity. In this case we have two linked problems, and we don't care about the "how" at all so long as its within framework requirements, we care about the end of "net zero carbon increase in the atmosphere" (or long term dealing with past emissions, though that'd require taxing other sources). This is exactly the sort of situation that's an ideal fit for using "market" (massively crowd sourced value apportionment utilizing capitalism) to optimize. On the one side is energy generation, which must directly pay per unit for the cost of dealing with their own emissions, and on the other side is finding the ways to most efficiently remove each unit and then collect the money. That would be expected to vary over time and by location, and to feedback into each other (and overall demand/efficiency) via the simple mechanism of price.