This might be more of a question of altitude and the ability to remove heat as a function of air density. The SR-71 has a cruising altitude of ~26km and the Concorde has a crusing altitude of ~18km. Of course a black coating makes it absorb more solar irradiance, but the designers would have taken this into account and ensured that energy balance favors a desirable temperature. The SR-71 had at most two passengers, so comfort cooling can be focused on the cockpit compared to a passenger plane. The Concorde on the other hand had a much larger passenger cabin which would lead to much higher parasitic cooling losses with a black skin.
This chart shows the decrease in density, but whether it's enough to justify a different approach to passive thermal control is hard to say:
The implication is that at high altitudes, heat generated inside the vehicle needs to be removed by a combination of convection and radiation. This is especially important for the Space Shuttle as it needs to maintain temperatures in the lack of atmosphere although the black tiles did not serve as the orbiter's primary radiators. They only needed to insulate the orbiter from aerodynamic forces and the high temperatures. While reentering the atmosphere, the heat comes from air compression in the atmosphere and the black tiles would have helped to re-radiate some of the heat away from the orbiter.