For what reason does one not recite facing excrement?

https://www.etzion.org.il/he/halakha/orach-chaim/prayer-and-blessings/keriyat-shema-problematic-situations-and-places-1

Let me clarify further. The problem is not a prohibition, but that such a recitation is worthless. Therefore, there is no room here to talk about a positive commandment overriding a prohibition. The positive commandment itself is not fulfilled in such a situation. As far as I remember, this is also not defined as a prohibition in the enumeration of the commandments (only in the context of the camp). From the law of the camp we derive a rule in the laws of Shema that one does not recite facing excrement. So there is no room here to talk about a positive commandment overriding a prohibition.

How do you know that what is derived from the camp is that such a recitation is worthless? With equal force I could say that what is derived from the camp is that it is forbidden to recite facing excrement, not that the recitation itself is worthless.

The prohibition is because the circumstances invalidate the recitation. There is no point in reciting the Shema in an invalid way.
There is a difference between reciting the Shema at the cost of an unrelated prohibition, where one could then discuss a positive commandment overriding a prohibition, and reciting the Shema in a way that Jewish law itself said not to do. It is like eating in an invalid sukkah because I have no other sukkah.
For example, suppose there is an opinion that a positive commandment overrides a prohibition even when both can be fulfilled. According to that opinion, the prohibition against reciting the Shema facing excrement would not exist at all. It would be emptied of all content.

Why should the circumstances invalidate the recitation? I’m trying to understand how you infer from the sources that when the Sages say one may not recite under such-and-such circumstances, that means this is part of the definition of the recitation, rather than there being a prohibition against reciting in that way.

How do you understand that eating in an invalid sukkah is not sukkah-eating, rather than being a prohibition? Reciting the Shema is, in itself, a commandment. So it stands to reason that when they forbid reciting facing excrement, it is not just some prohibition, but a definition of how one recites Shema. It seems completely straightforward to me.

If there were a general prohibition against standing facing excrement, and it conflicted with the commandment of reciting the Shema, then we would have here either a commandment that comes through a transgression or a positive commandment overriding a prohibition. But there is no such general prohibition. This is a detail that defines the commandment of reciting the Shema.

As for sukkah, the Sages did not say that it is forbidden to eat in a sukkah with two walls; rather, they said that a sukkah must have three walls. And sukkah, unlike recitation, is something that needs a definition, whereas recitation is something understood on its own. It is like eating not needing a definition of what counts as eating. Also, why is there any need to bring sources like “and your camp shall be holy,” or “that He not see in you anything indecent,” in order to define the Shema? If, regarding sukkah, they brought an external source saying that it is forbidden to eat in a sukkah of such-and-such a type—for example, suppose it were forbidden to eat in a sukkah that has excrement in it by the rule of “that He not see in you anything indecent”—then there too I would say that this does not define what a valid sukkah is, but rather that there is another law that forbids eating in a sukkah that has excrement in it, and the usual Torah rules of conflicting obligations would apply to that law.

I saw that Peninei Halakha writes as follows:

If one recited the Shema or prayed within four cubits of excrement, he has not fulfilled his obligation, and he must go back and recite the Shema and pray again. But if he did not know that there was excrement there: if it is a place where there is a reasonable concern that excrement might be there, and he was negligent and did not check the cleanliness of the place before praying, he has not fulfilled his obligation. But if it is a place where excrement is not likely to be present, then after the fact he has fulfilled his obligation (Shulchan Arukh 76:8; Mishnah Berurah 31).

Regarding the last clause, which says that after the fact he has fulfilled his obligation, it would seem not to fit with what you said, that this is part of the definition of reciting the Shema. Just as with sukkah, if someone did not know that he ate in a sukkah with two walls, it is obvious that he did not fulfill the commandment of sukkah even after the fact.

The commandment always rests on the person, so the definitions can always be formulated in relation to the person. One can say that sitting in a sukkah without walls is not sukkah-sitting. Or one can say that it is not a sukkah. In Shema there is no object involved, so there the invalidity is in the person’s act (the recitation). But it is the same thing.
As I explained, there is no prohibition against standing facing excrement. It is a rule within the laws of Shema.
The fact that this is derived from the law of the camp is only a derivation that such a state is not one of cleanliness (holiness), and Shema requires cleanliness. So that says nothing for our purposes.
And finally, Rabbi Melamed says exactly what I wrote: that it invalidates the recitation (and this is not because of a commandment that comes through a transgression, but from the laws of Shema themselves). The fact that sometimes this does not prevent fulfillment after the fact proves nothing. It means the recitation is problematic, but not invalid. And still, it is a rule in the recitation, not an independent prohibition.

Thank you very much to the Rabbi and to Oren!