> Pigs I may be wrong about. They're clearly more similar than dogs, cats and cattle. In embryonic development, for sure, and I remember that from dissecting a fetal pig. And there is the saying that embryology recapitulates phylogeny. But a quick search didn't come up with any DNA-based phylogenetics.

The saying exists, but it is not based in anything other than wishful thinking. Ontogeny does not recapitulate phylogeny; that saying is the tongue map of evolutionary analysis.

https://en.wikipedia.org/wiki/Recapitulation_theory

> the shortcomings of the theory had been recognized by the early 20th century, and it had been relegated to "biological mythology" by the mid-20th century. This did not prevent its propagation into the 21st century in a variety of biology textbooks.

If you believe the page on euarchontoglires, you can see that pigs cannot be more closely related to humans than dogs, cats, and cattle, because all four of those human-other pairings have exactly the same common ancestor, the common ancestor of the euarchontoglires (the tree branch containing humans) and the laurasiatheria (the branch containing pigs, cattle, cats, and dogs). Any closer "similarity" between pigs and humans in specific is not driven by their evolutionary relationship.

> DNA-based classification is really the only way to get objective phylogenetic trees. And as we get them, we learn that lots of classical taxonomy is just wrong.

DNA-based classification is not conceptually different from morphology-based classification. They are both directly premised on the model of "descent with modification". DNA-based classification is generally superior because there is more information in the DNA than in the morphology. (Or at least, we recognize more information in the DNA than we do in the morphology.) But morphology-based classification did nearly 100% of the work of establishing modern taxonomy, and overwhelmingly DNA-based analyses confirm the existing morphology-based conclusions.

> traditional taxonomy is bad at determining how related species are

This is true in the sense of assigning a cardinal number to the relationship. ("Diverged 80mya.") It is false if your goal is only to assign an ordinal number. ("Humans are closely related to baboons, distantly related to rabbits, and remotely related to elephants.") The same objection applies to DNA-based taxonomy; we get cardinal numbers by assuming a particular constant-rate molecular clock, but the raw conclusions of DNA analysis are entirely ordinal.

> as we get them, we learn that lots of classical taxonomy is just wrong.

(quoting again to respond along another topic)

Notwithstanding the overall massive success of traditional taxonomy, there are some obvious weak points. They tend to derive from philosophical beliefs of the biologists rather than from failures of the morphology-comparing model. Two major biases that I'm aware of are the historical belief that taxa at the same level should include roughly the same number of extant species -- this was featured in a "teach the controversy" sidebar in a college biology textbook of my mother's -- and the stronger view that genera should contain more than a single species. Since the groupings prompted by these beliefs were not even intended to "cleave reality at the joints", it's not so surprising that they suffered under more objective analysis.