> Wouldn't the equilibrium temperature be both a function of energy in and radiation rate (which is also a function of temperature?)
Exactly, and since we're not changing the energy in (that energy is going to reach us whether we pump it up or not), and thus the equilibrium does not change.
Where else can all that generated energy go other than up?
My explanation would be correct if the earth is in equilibrium and there is no primordial heat left and all heat comes from radioactive decay, but surprisingly (to me at least) that is not the case. About 1/2 of the heat flow from the core to the surface comes from primordial heat.
I'd say solar will also add some heat to the atmosphere because they reflect less sunlight than the surface.
Of course, in the big scheme of things this is all pretty irrelevant unless we keep on exponentially increase our energy usage. The greenhouse effect of CO2 is a few orders of magnitude higher.