...the actual historical linguists firmly dismiss them as unsound, even though they have no answers of their own to offer instead. [7] It’s an example of a commendable stand against seductive nonsense.
...
What is important in this context is that the mainstream consensus has never accepted any such estimates into its body of established knowledge, even though they provide superficially plausible answers to tantalizing questions.
...
Do you know of any concrete breakthroughs in historical linguistics achieved by studying trends in verb conjugation?

I wasn't arguing they aren't dismissed, just that perhaps they shouldn't be. Arguing that there are no accepted breakthroughs is weak evidence against that, if the method is rejected by the mainstream the strongest favorable evidence we might have expected to see would be theories proven by other methods, theories found in hypothesis space by using the questionable method. I don't know if this is even possible in linguistics.

It dredges the English data to derive a rule that I'd bet would be falsified if one were to study other languages.

The scope of the rule can be the wrongest part, and it would still be useful. The rule as stated might be specific to Germanic languages but be an instantiation of a more general concept.

The counterexample you spent the most words describing would be the typical strongest sort to give to a hypothesis in that you described the most extreme cases of irregulars becoming regular. But the hypothesis of the paper, read charitably at least, is not challenged by it. It allows for "the 1st person singular of this irregular conjugation has spread to nearly all verbs, and is now the regular one," as it's about the rate of change in conjugation once a regular rule takes over and begins spreading. At some point in Croatian the irregular conjugation had enough momentum to fit under a moderately changed version of the hypothesis.

The counterexamples you only hinted at would be stronger. Are there coexisting regular rules of conjugation in other Slavic languages, with irregulars assimilated variously into one or another regular rule? If so, I think that wouldn't challenge the thrust of the argument unless verbs changed between rules.

My entirely uninformed perception of the verb-based method is that it has low sensitivity but isn't invalid compared to the other linguistic methods.