I know for sure painting the tanks won't do anything because those parts aren't intended to shed heat anyway, they are just tanks to hold the coolant. And just an assumption but unless you manage to fully coat all of the fins and tubes through the entire thickness of the core (not easy) I highly doubt it will affect the heat transfer ability. "Radiator" is a misleading name anyway, they don't work off radiation they work off convection. Black vs. white surfaces only affect radiative heat transfer, like from an old-school cast iron stove that sits in the middle of a room. You feel the heat coming off it even if the air isn't moving around.
I was worried about painting my radiator at first, I was looking for special "radiator paint" then I did a common-sense reality check and just used some simple Krylon combo black paint+primer (less coats needed without a separate primer). Could still be a real thing though I don't have evidence to say it's not true.
