The surface temperature of the Sun has to be about 6000 K to make the interior structure work. Given that, the Sun would have to be a blackbody at nearly that effective temperature, no matter what the thermalizing opacity was. All that can change is whether the actual temperature at the surface is pretty much the same as the "effective" temperature, and that requires some kind of significant source of thermalizing opacity-- but what the source is does not matter terribly much ("thermalizing" opacity is opacity that is capable of destroying pre-existing photons and creating new ones based entirely on the local temperature).

Having said that, it seems your question remains, what is the thermalizing opacity that serves this purpose? The answer is, "H minus opacity". What this means is, there are a few neutral H atoms near the Sun's surface that have acquired a kind of "hitch-hiker"-- an additional electron in one very weakly bound state that makes the H a negative ion. That electron is easily knocked off the atom, but when a photon does it, that destroys the photon. The energy goes into free electrons, which can later make new H minus atoms-- in turn emitting new photons in a way that depends on the temperature of that bath of free electrons. So you have just what you need-- thermalizing opacity that is capable of acting over a wide range of frequencies (because photons of almost any energy can knock the electrons off, and can be created when electrons that have various amounts of thermal energy are captured in that bound state).