Two things to say here:

(1) The view articulated in that answer, that the Second Law only applies to systems that are genuinely closed, would render the Law empirically useless. There are no systems of this sort, except for the entire universe. But we appeal to the Second Law all the time to account for the time-directedness of systems that aren't completely closed (such as ice melting in a glass of water, or gas spreading through a room). We're really working with an approximate sense of closure, one that allows us to describe reasonably insulated systems as closed (with the denotation of "reasonably" depending on context), even though technically they are exchanging some amount of energy with their environments. If we go by the standards in that post, then yes, no system we observe would be governed by the Second Law. But by the same token, the "system plus observer" supersystem wouldn't be governed by the Second Law either, since this supersystem isn't closed. So then I don't see the point of defending the Second Law by including the observer in the system.

(2) The "begging the question" charge I raised in my post is not merely hypothetical. Shalizi is genuinely skeptical of Landauer's principle, the claim that information erasure must have an entropic cost. So invoking Landauer's principle won't fly against him. I think the right response to the sort of problems he raises with the principle (best captured in the John Norton paper linked in his post) is a view of the sort I recommend above. I'd probably need to say a lot more to make this obvious, but I won't unless you're specifically interested.

ETA: Also worth noting: All competent defenses of Landauer's principle that I have read assume that the observer is governed by the Second Law. The usual argument involves pointing out that erasure involves a reduction of the information theoretic entropy of the data stored by the observer. Since the Second Law holds, this reduction of entropy must be compensated by an increase of entropy in the non-information-bearing degrees of freedom, which usually amounts to the observer releasing heat into the environment. But if we go by the reasoning in the answer to which you link, we have no warrant for assuming the observer is governed by the Second Law unless the observer counts as a genuinely closed system. Of course, no actual observer would qualify. So the poster's own reasoning undermines his appeal to Landauer's principle.