A) It's an excellent idealization of the appearance of the sun, and not intended as a dynamic model of the interior.
B) None of this has anything to do with the solar system being an OPEN system ! Talk about nit picking ! I was pointing out that IF the solar system were closed, it would be uniformly hot. Then you say, oh no, much hotter than that! and the solar surface would disappear ...
So how is this supposed to support your completely untenable assertion that the solar system is thermodynamically CLOSED ????
As for closed system: with respect to the question at hand, the OVERWHELMING energy of the sun's radiation is completely irrelevant to this discussion, because it does not fall on the surface of the earth, and even most of what does isn't absorbed. Or do you think that the number of photons that reaches the Andromeda Galaxy somehow impinges on a discussion of Entropy and evolution on Earth? [Hint: It does not.]
Finally, with respect to the absorption/emission of radiation question. Of course I am not talking about the Entropy of emission and absorption of single photons, each of which with a properly understood quantum mechanical definition of entropy is certainly governed by the Second Law. The radiation field of the sun gains Entropy as it goes from T(high) to T(Low) as you note. [But, Please! You aren't talking to a Creationist here] Evolutionary processes lose entropy which must come from this field. Here is an interesting paper [How much Entropy Reduction Does Evolution Require?] which based on some very liberal estimates suggests just how large that Entropy decrease could be. In trying to find it again, I pleasantly note that it's been improved [Evolution and The Second Law of Thermodynamics ] since I first read it.
You will perhaps be disappointed to note that in both this paper and Styer's original work, the total solar radiation escaping into space is not considered. Note also that it discusses an Entropy decrease; your signum is, as I pointed out, incorrect.