Maybe I was not enough clear for make you underestand my post, Giant Panda. Probably this will make you underestand better. Most, if not almost all, recent taxonomic changes at any taxon level are based exclusively in ADN molecular analysis. While in the past years, the taxonomy was based in much more complete studies regarding morphological, biometrical, biochemical, ethological, structural, etc. (and also molecular analysis as a "helping hand", that is as it should be, but not basing changes exclusively on it). Basically, a molecular analysis not always give reliable results and a PCR can be made by anybody with informatic knowledge just using a program, without need of scientific knowledge. As you said, I'm not much fond on the details of this but I know by more knowlegdeable people than me, that the former science is much more reliable, and I think that molecular analysis can give different results with cells of different parts of the animal, for example. Anyway, the result is that one learn in memory a good-sense looking taxonomic schemes (without being fooled by evolutive convergence, etc) and nowadays most of these schemes are changed by others that looks unappropiate or almost ridiculous sometimes. With your permission, I will take your nickname as an hypothetical example. Classic science changed the taxonomy of giant panda over the time, puting it first with bears, after with racoons, after with red panda (as they also have a "sixth finger") and nowadays with bears again (that seems the most logic due to the almost identical morphology). Imagine that one day, any person make a PCR of a giant panda and decide that actually is not in the order Carn¡vora but a giant shrew (that is the kind of things that molecular analysis do... if you don't believe, just see how whales and giraffes are now in the same order, while others suggest that falcons aren't raptors but parents of parrots, while bats are not an order but two...). Would you believe in it?
This is something of a non-debate, because molecular data has been overwhelmingly vindicated. No-one is trusting to "blind faith" or discarding "all the science that has been ever", but the point of science is that it doesn't stop at an arbitrary point in the 1970s because you felt comfortable with that arrangement. When morphology and genetics are used to construct phylogenies on the same groups independently, the latter almost always gives higher resolution and support.
Genetics carries an array of advantages, but perhaps most important is the vast datasets available (ie. a whole genome), compared to relatively few morphological characters. A 2006 metaanalysis found that genetic phylogenies typically used 10x more parsimony-informative characters and this is increasing year-on-year.
Re-sampling techniques, eg. bootstrapping, also allow us to establish the statistical support of a topology. With statistical analysis, we can identify not just the most probable tree, but also its probability compared to competing trees. Morphology doesn't work like that, because it uses fewer characters. That's great if those characters do define valid taxa (eg. angiosperm carpels), but not if they don't (eg. totipalmate Pelecaniformes). So whilst you say morphology is more "reliable", it quite simply isn't. The datasets available to molecular analysis make it, in general, a far more reliable tool.
You also say that morphology should be used provided we're careful to ignore convergent evolution, but it wouldn't be an issue if we always knew what to ignore. Genetic data face similar problems, of course, but much of the genome isn't under direct selection. Characters known to change at a fixed rate purely as a result of drift are a morphologist's dream, but a geneticist's reality.
And I'm not entirely sure why you're so disdainful of PCR, but ease and speed are both major practical benefits of genetics. In science, difficulty doesn't mean efficacy.
Of course there are issues with genetic data, some endemic (contamination, long-branch attraction) and some not (parallel evolution, insufficient sampling), but increasingly sophisticated techniques are being developed to overcome these. As the example you gave, sequencing different parts of the genome can certainly produce conflicting results. Three points, however: 1) Sampling a range of loci massively reduces the problem; 2) Morphology isn't immune to this, either; and 3) Conflicting signals within the genome can prove informative, for example demonstrating historic admixture.
I find your panda thought experiment a little silly. It's like me asking whether you'd believe the morphological evidence placing giant pandas in the Soricomorpha if they were discovered to be 5cm long, venomous, and have a pointed nose. They don't have any of those traits, so the question is meaningless. Cetartiodactyla, by contrast, is a very well supported clade, based on extensive molecular data and "classic" palaeontology. Incidentally, a major line of evidence which suggested relatedness between the pandas was their karyotype – a crude genetic method. Without improved genetic analyses, however, my namesake would still be flip-flopping between the ursids and procyonids.
As with anything, not all genetic studies are equal, but it's a fantasy to pretend that morphology didn't also give rise to disputes. Molecular data has put us in a far better position to resolve these.
so your final example is incorrect, as this is a hypothesis which:
