Are there any explicit examples of poor methodology or application of the historical comparative method that most, if not all, can agree on?
In other words, you are asking for the fundamentals of the comparative method. There are books written on the matter. You ask for an example of somebody not following them.
long range etymology
Most public opinions agree that the Altaic language family was highly suspect. Altaic remains in use as the label for a group of Languages with obvious similarities, but the comparisons to Korean and Japanese for example are disregarded. The critique centers on missing hypothesis of a system of total sound correspondances, often spourious semantic comparisons, that remain the authors' secret, and preference for competing theories. It's a convolute I wouldn't touch with a nine-foot pole, so I cannot judge its merrit. But the inflamatory tone of the most vocal critiques has given them a bad rep, too. It's hard to tell who's the sinner.
Nostratic, Proto-Human and other long-range studies hit a similar vein. Few neutral bystanders in the literature offer encouragement. Sergei Starostin and, earlier, Vladislav Illich-Svitych have died at a relatively young age, and it is not clear to me whether that was just a euphemism for abandoning their carreers admitting defeat (no disrespect; take incredulity as a stage of grief, if you will), or really just unfortunate accidents (a heart attack in Starostin's case; Svytich ws involved in a car accident), or the consequence of manic, self destructive behaviour. At any rate, I try to bear the common warning in mind, that superficial similarities can be random chance at an overwhelming rate, that cannot be sorted out trivially.
However, statistical methods are supposed to counter this problem. Those that I'd seen criticized focus on the very limited input of 200 item swadesh lists (perhaps as a test). Ruhlen et al have produced a very convincing shortlist and, drawing on the StarLing database, amassed a seizable list of what might be thousands of cognates, despite the perhaps erroneous reconstructions. That is, they show consistent semantics, relaxing the constraints on the Phonology instead. The mistake then is to jump to conclusions about genetic relations, without cross-references from the other Anthropological fields.
Otherwise, many etymological dictionaries, mostly older works, have been called the product of "insanity", "worthless" and "nonsense", whether Hebrew, English or Turkish, either because they do not offer a sound methodology (Whom Anatoli Liberman dubbed "Monomaniacs"; Comparing a single language with itself, or only one other that's supposedly the gospel of truth), or because they show disregard for the prior research, if not ignorance.