The second part of the post describes a "valve" which actually lets some radiation to escape, so is worth examining. However, in summary this is another example of a poorly labelled diagram, which in this case seems to have confused the author himself;
I urge you to notice that the valve's efficiency doesn't actually matter, either, because physical laws are violated even in a modest case. In some sense, in fact, the crimes get worse. For instance, let's install a 20% valve, so that 80% of the infrared escapes and 20% back-radiates.
In this case, 0.8 exits while 0.2 is "retained" by the surface. But 0.2 also radiates back to the surface, so it gains 0.4 in total (again, as a minimum: further back-radiation effects must arbitrarily be halted). In other words, even when the oft-mentioned "net flow" favors the outward movement of thermal energy (a modeling effort to satisfy the Second Law), the alleged heating effect still contradicts the First Law because you're getting more energy than you put in. Any furnace manufacturer would eagerly exploit such a loophole in the law if it existed.
"In this case, 0.8 exits while 0.2 is 'retained' by the surface" - yet the upward red arrow at the top reads 0.8, meaning that the red arrow below it must represent 1 unit in order for the "valve" to have absorbed 0.2 units. Not a good start, since if the surface "retains" 0.2 it can only radiate 0.8 with a corresponding 0.64 exiting the "valve". The 0.8 shown escaping plus the 0.2 "retained " by the surface plus the 0.2 absorbed by the valve add up to 1.2 and not 1. Earlier in his post he said "You can't obtain more energy than you put in", yet he's just done it! Apart from this fundamental error, if the surface retains a net anything the system can never reach equilibrium - it has built-in positive feedback from the word go. He says earlier "for every unit of sunlight going in", so if for every unit going in 0.2 is "retained" the surface will continue to heat up, and thus far in the process the the "valve" has played no part in any "feedback". So the diagram doesn't actually represent what the description says it does. In reality the surface never does actually radiate 1 unit, as implied in the diagram, unless the"valve" is initially set to allow all radiation through, and then activated. All that's needed (once again, as in "A Simpe Solution") is to consider the energy actually lost from the system to get a properly thought-out analogy or "model".
At equilibrium, 1 unit escapes through the valve, which means that the surface is radiating 1/0.8 = 1.25, and the "valve" is absorbing and re-radiating 0.25 downwards. But where does that "extra" energy come from? It comes of course from energy absorbed during the equilibrium process. initially the surface acquires 1 unit, absorbs that energy and heats up until it radiates 1 unit. The "valve" absorbs 0.2 of that, and it heats up until it radiates the same down. The process continues with diminishing increments - the 0.2 is absorbed by the surface, increasing its heat content until it radiates 1.2 units, with 0.96 escaping and a total of 0.24 absorbed by the "valve" and so on. Of course, the process doesn't actually have discrete steps - it's smooth and continuous. The heat content of surface and valve increase until radiation in and out of the system balances and equilibrium is obtained. No physical laws are violated, and "furnace maufacturers" are well aware that it takes time for their products to heat up to operating temperature when turned on.
There's the nub of the matter - the "Slayer's" misunderstanding of the difference between heat flow amd heat content. The surface of the Earth and the atmosphere don't radiate because they are absorbing radiation, they radiate because they contain heat. Radiation from the sun merely tops up the heat content. At night there is no radiation from the sun, yet of course the Earth/atmosphere system cotinues to radiate heat to space, slowly cooling. It's surely a simple concept to grasp, yet it appears to elude some.