I've always figured vestigial organs were a stronger argument against intelligent design than stuff like circuitous nerves that, while fulfilling some purpose, are thought to be "bad" designs. For example, whales have these useless hip bones floating around, separate from the rest of their skeleton. It doesn't really make sense to have these pointless structures there if whales were designed from the outset to be best for their niche, because, I mean, they're pointless. But it makes more sense if you consider a stupider evolutionary explanation whereby there was no pressure to select against these unused hipbones, but there was pressure against the limbs that articulated with them, so the hips just stayed when the legs went away.
Even then, this seems kind of weak as a counter-example (at least to versions of ID that accept evolution), because there's no way to definitively say that some intelligent designer didn't intend for those hip bones to remain present because whales might need to make use of them later in response to future selection pressure. Meanwhile, there's evidence of parallel evolution (e.g., similarities in the eyes of octopuses and humans) and the notion that many structures are "optimal" for their function, both of which could intuitively be explained by some underlying design principles guiding it all.
The real beef I have with ID (even the versions that accept evolution) is that it just kinda reeks of magical thinking and doesn't seem to exist for any purpose outside of framing things in a way to validate religious beliefs. It just smells fishy to assume the necessity of some transcendent, intelligent being whose main (and apparently sole) influence on the world is optimizing each plant/animal/prokaryote etc. for its ecological niche, largely because it raises a ton of questions about the nature of this being that, while necessary to make predictions, can't really be tested due to that pesky/convenient "transcendent" property.
Consider the Darwinian explanation that you consider to be "stupider". Why do I as an empiricist accept this explanation over intelligent design? Well, one knows from aggregated experiences of the scientific community, that is from observations and experiments, that Darwinian evolution is universally operative through the biological world: no population is not static relative to its environment. Indeed, any population with heritable genetic variation in fitness (defined relative to the members of the population in a specific ecological niche) would undergo natural selection, and one can infer that many extant biological traits of a given species are the cumulative consequences of the selection process that the ancestors of the population experienced. You are correct that there is no definite way exclude intelligent design, and epistemically, one has to make an inductive assumption, based upon scientific experience, that natural selection operates in the past on the ancestor of whales. This is a key assumption. and there is no necessary reason why would it be true. Yet, one justifies this assumption on the solely basis of experience, as sensory experience, to an empiricist, is the primary means one attains knowledge about the world, even though one cannot have any direct experience about the past evolutionary history of a species.
Evolution is indeed a historical question, and the explanation may be seen as quite vacuous, since it does not list the specific selection pressures that the whale lineage historically experience, as it just appeals to the general processes of natural selection and the absence of selection pressure to remove the hip bones. Similarly one can also ask why does one accept kinetic theory as a reasonable model of the molecular behavior of gas molecules. Note that kinetic theory is an "explanation" for observed macroscopic phenomenon. It is not merely a recitation of empirical relations such as the proportionality of temperature and volume of a gas under the same pressure (Charles's law); it explains why Charles's law should manifest itself in the light of our understanding of the world. Kinetic theory makes a few reasonable assumptions about their behavior -- such as they exhibit Newtonian mechanical behavior, are point particles that take up no volume, and have no intermolecular interactions -- that one cannot directly experience. (These assumption are currently regarded by scientists as literally false, but we regard them as practically reasonable because the kinetic theory can model the behavior of gas under most conditions.) For one's crude personal senses, one cannot observe these putative properties of gas molecules.
But how does one accept that kinetic theory provides a reasonably accurate account about the behavior of gas molecules? The epistemic justification of kinetic theory is based on abductive reasoning, since one do not possess knowledge
a priori the behavior gas molecules nor could we directly verify these properties by directly observing them. Kinetic theory makes non-trivial predictions based on these assumptions about more observable macroscopic phenomena. Essentially these predictions are a translation of the microscopic behavior of the gas molecules to the macroscopic realm. When these predictions are observed in a scientific experiment, it supports but does not prove, that the underlying reason for the observed phenomena is the existence of gas molecules that exhibit Newtonian mechanics. This is actually a formal fallacy called "affirming the consequent", since even though the observation conformed to the expectations of kinetic theory, it does not exclude other possible models that are compatible with the observations. Since there are numerous instances (that outnumber the exceptions) of gases that exhibit behavior that complies with kinetic theory, one can induce that most gas molecules (even those not subject to the attentive eyes of scientists) exhibit the microscopic behavior ascribe to them by kinetic theory. Indeed, an epistemic inquiry of the nature of scientific knowledge does not provide a confutation of philosophical skepticism as there remains some philosophical (even though they are not scientific or practical) grounds for skepticism even for the most established scientific theories.
Unlike the past selection pressures that acted upon whales, scientists can get a fairly quick "confirmation" of kinetic theory by carefully conducting a scientific experiment with the appropriate instruments. Scientists cannot reconstruct conditions to investigate rigorously the evolution of the whales. Charles Darwin formulated his theory of natural selection on the basis of his experiences as a naturalist, such as his Voyage on the
Beagle when he experience the diverse flora and fauna of the world, and he noticed interesting regularities through nature, such as variance of the beaks of the various birds of the Galapagos as adaptions for the specific habitat of their respective island. Due to their geographic proximity and physiological similarity, he thought that these species of birds originated from a similar common ancestor whose descendents migrated different islands and subsequently were subject to different selection pressures. To support his principle of natural selection operating in nature, Darwin noted similarities from the science of selection domestic breeding as the initial population had variation in the trait of interest, and in subsequent generations, the population exhibited more of the desirable trait because members of the population that had more of the desired trait were artificially granted more favorable mating conditions and had more offspring. Based on the more observable conditions of domestic breeding, Darwin reasoned that an analogous process operated in nature where it could not be scrutinized as easily by scientists due to the scope of the biosphere and the time required for the necessary adaptions, because there would also exist population variation and fitness differences. From these observations, Darwin made the more grand and universal conclusion that many of the traits of all biological species, including and especially humans, were derived from the process of "blind" natural selection. The key point that Darwinian evolution is not based on the direct experience of the various histories of the biological species of Earth, but it is constructed from inductive inferences in the light of human experience.
Now, the next question is that why should an empiricist reject teleological explanations. Darwin supported his theory by adducing his personal experiences and the documented observations by other naturalists; Darwin and other naturalists had sense experience of natural relations (such as the geographic proximity of the birds of the Galapagos), and Darwin's experiences from his personal observations and reading made powerful impressions on his mind that led him to formulate the theory of natural selection. In other words, the theory of natural selection was ultimately the fruit of human experience, not from derivable from
a priori reasoning alone. And thus the theory of natural selection is thus epistemically subjected to the limitations and follies of human experience. Now do humans have strong sense impressions or ideas derived from these impressions from observing the world that one can credit specific traits of natural objects to be the work of an intelligent designer? Certainly, one does not observe the influence of a supernatural designer intervening in the realm of biology in any way to produce intricate traits, and therefore one cannot acquire any sense impressions about the designer's work on biological creatures. To the contrary, we know houses are intelligently designed objects because we have observed the processes of house construction, and saw that it involves the plans of an architect and human laborers to assemble it. We then can appeal to human experience and the principle of induction to come to the conclusion that a particular house involved human design, even though one does not possess direct experience of that house's origins. But we cannot do this for something such as the bacterial flagellum due to its dissimilarities with human constructed objects; the only "evidence" of the intelligent design of the flagellum is one's lack of experience about the flagellum's origins and insufficiency of one's mind to conceive of a plausible naturalistic origin. Because of the insufficient sensory impressions and ideas about the activity of an intelligent designer, one cannot credible promote a theory of intelligent design on the basis of human experience. Scientists reject intelligent design and supernaturalism in the sciences in general, not due to some
a priori commitment to naturalism, but due to the success of naturalism in science.