Furthermore, one must remember that 5777 Kelvins is an effective temperature, a best fit blackbody to an actual thermal emission that is a superposition of blackbody emission curves that are generated at different depths in the photosphere. Limb observations of the sun make it possible to retrieve the temperature structure of the photosphere as a function of depth, in much the same way as limb observations of Earth's atmosphere by satellites allows us to retrieve temperature profiles for the Earth's atmosphere (see, e.g.,
Solar Astrophysics by Peter Foukal, Wiley-VCh 2004, chapter 5: "The photosphere";
The Observation and Analysis of Stellar Photospheres by David Gray, Cambridge University Press 2005, 3rd edition). The temperature at the lowest level we can determine is 9400 Kelvins. We don't see much of that on Earth, because of the opacity of the overlying layers. But your iron surface is pretty much hugging the 9400 Kelvin base of the photosphere. To the best of my knowledge, the highest boiling point for any element is
Rhenium, which boils at 5869 Kelvins, so no known element can survive as a solid or even as a liquid at the temperature found at the base of the photosphere.
The iron surface of the sun is thermodynamic toast, and "thermalize" is a pleasant fiction that bears no resemblance to the physics of this universe.