But consider that the total thermal energy of the Sun is negligible compared to the total energy flux. And that thermal conduction is way too slow, anyway.
Really, do the math. Take the sun's mass, and multiply by average absolute temperature and the specific heat of hydrogen. That'll be a huge number, but far less than the sun's energy output.
You could also compare energy output with what's possible through thermal conduction, based on the sun's radius, its density at various depths, and the heat transfer coefficient of hydrogen.
But consider that the total thermal energy of the Sun is negligible compared to the total energy flux. And that thermal conduction is way too slow, anyway.