Answer

I’m not sure I understood you. The Talmud says that one could have derived it through a common denominator with regard to stones, and the Torah wrote a verse so that we would not do so. I take it that you are asking: why not derive it despite the similarity here (there is no human intervention), meaning, what is the reason for the verse saying not to derive, if in fact the stones really are similar to the common denominator of the source cases? Did I understand your question correctly? If that is the question, I would suggest two answers, with a practical difference between them.
In the straightforward reading, it seems that the verse says that despite the similarity, we do not derive here for some other reason (perhaps there is a hidden refutation that we would not have noticed). But there is another possibility, and I am more inclined toward it.
We find in several places the refutation of “a stricter aspect,” meaning: what can be said of the two source cases, where in one there is a stricter aspect X and in the other there is a stricter aspect Y, which does not exist in the target case? (Of course, here in our passage this is actually a lighter aspect and not a stricter one, since we are deriving a leniency—permission for benefit—but that does not change the logic.) See, for example, Ketubot 32a (and Tosafot there below), and Makkot 4a, with the lengthy discussion in the Ritva there. Tosafot and Ritva there ask that if we can refute a common-denominator derivation by saying “stricter aspect,” then you have eliminated the whole institution of common denominator from the world, because each of the source cases always has some stricter aspect. Were it not so, these would be two verses that come as one and therefore do not teach. They gave various somewhat forced answers. I once suggested what seems to me a very logical explanation for this (I wrote it in several places, if you want).
The usual assumption in a common denominator is that the special characteristics of each source case are not what causes the law: case B proves that X is irrelevant, and case A proves that Y is irrelevant. It follows that presumably their shared characteristic, Z, is the relevant one—that is, it is what causes the law in question—and that characteristic is also present in the target case. Why is the explanation that attributes the law to Z preferable to an alternative explanation that attributes it to the conjunction (X or Y)? Because of Ockham’s razor: it is preferable to attribute the law to one characteristic rather than two. It is a simpler theory. (Every scientific generalization is based on this logic: why assume that mass is what causes an object to fall to the earth, rather than its rectangular, circular, or triangular shape, if those too were among the shapes of the objects I observed?)
That is the regular logic of a common denominator. From here you can understand that one must distinguish between halakhic characteristics and factual characteristics. If X and Y are factual, then the refutation of “stricter aspect” is really not relevant. But if these are halakhic characteristics (stricter laws that exist in case A and case B), then one could say that the two source cases share one factual characteristic (which does not exist in the target case) that is expressed halakhically in a different way in the two source cases. In other words, they have two different stricter laws, but the same factual characteristic underlies them both and causes those differing laws in different contexts. If so, perhaps that underlying characteristic is what causes the law we are trying to derive, and therefore one cannot derive from them to the target case, because that common underlying characteristic is not necessarily present in the target case, since it is not stringent. That is the refutation of “stricter aspect.” And in fact, if you check throughout the Talmud, the refutation of “stricter aspect” is raised only regarding halakhic characteristics, never regarding factual characteristics.
Let us now return to the Avodah Zarah passage. First of all, there we are obviously dealing with factual characteristics (attached to the ground, and living creature), and therefore seemingly the refutation of “stricter aspect” does not apply there. So in principle we really would derive there by common denominator. My claim is that the verse comes to say that even though these are factual characteristics, one should nevertheless apply the refutation of “stricter aspect.” That is, the verse reveals that here the law—the leniency of permission for benefit—is caused by each of the two characteristics separately (either being attached to the ground or being a living creature), and not by some common denominator shared by both of them (namely, the absence of human intervention). In other words, here one should not apply Ockham’s razor. As a result, this leniency exists only in the two source cases, and one cannot derive it to the target case (the stones), because they are neither attached to the ground nor living creatures.
And the practical difference would be for something else in which there is no human intervention, but which is neither a living creature nor attached to the ground, yet is not a stone. Regarding such an object, there is no verse preventing a derivation that it is permitted for benefit, because the verse speaks about stones—but even so, it would not be permitted. That is because the verse teaches that permission for benefit applies only to something that is a living creature or something attached to the ground, and not to every case lacking human intervention. By contrast, if we understood the Talmud in its simple sense—that the verse comes to teach a specific stringency regarding stones—then the verse would merely be teaching that there is no permission for benefit only with regard to stones, and that is all. Everything else with no human intervention would be derived by common denominator from the two source cases and would be permitted. I am not even sure that logically we would need the condition of no human intervention, because once we reject the common denominator, the cause is no longer the shared aspect (the absence of human intervention), but rather each of the two unique characteristics of the source cases.
In my opinion, this explanation is borne out precisely by Rashi’s wording there, where he wrote:
“This is why it says, ‘you shall utterly detest it’—with regard to idolatry, to include these cases, that even though they could have been derived legally by a common denominator toward permission, as we said, you should not derive them toward permission. And this emerges from the implication of the verse, which means: all derivations that you expound regarding idolatry, do not expound them except toward detestation and prohibition, and you have no right to permit in it anything except what Scripture explicitly permitted.”
It is not clear why he adds that the novelty is only in what was explicitly permitted. On the contrary, everything—that is, at least everything lacking human intervention; as noted, I am not sure that is needed at all—should be permitted, since everything is derived by common denominator from what was explicitly permitted, and only stones were excluded, since regarding them the verse introduced that one does not make a common denominator. It must be that he understood it as I suggested: the verse comes to teach that permission for benefit was stated only regarding the two source cases (a living creature or something attached to the ground), and not regarding other things, even if they are not stones.