| Wouldn't the heatsinks need to have very high temperature conductivity?
I don't think so. If you were to build /very tall/ tubes arching up into the upper atmosphere and cycle water through them (warm water up, cooling, and them travelling back down the other side of the arch), you could easily make them out of insulated concrete and still have to worry about freezing rather than lack of cooling. Of course, the problem then becomes what material can build this sort of structure, balancing height with throughput to prevent freezing.
But over long time periods, I think that you would also have to worry about the heat sinks evaporating the upper levels of the atmosphere.
If you transfer the heat to the atmosphere, it won't leave the Earth+atmosphere system, so the net effect will be zero. To actually cool the earth, you'd need to heat the atmosphere enough to make parts of it escape Earth's gravity. Aside from problematic effects on weather, this would be really hard because the upper atmosphere is very thin and so has low heat capacity and low heat conductance.
I was thinking about direct-radiation heatsinks: the inner part made of super-heat-conductive material that transports heat from ground level to a huge radiator fan...
Mine was to work tax policy to incentivize companies to make all their packaging shiny and white, incentivize people to litter, and disincentivize everybody from recycling.
My friend's was to use a giant rocket to push the earth farther away from the sun