My understanding is that the rigorous definition of convergent series was given by Cauchy and Gregory's (much greater) contribution was merely to hypothesize that only some series should be considered to have sums. This is an advance that I think is not so well connected to your suggestion.

Well, no, because it turns out that you have multiple notions of what it means to take the sum of the series. Abel summation would be one example. That said, I agree that not everything should be summable is a major insight, but even this came about as part of Cauchy et. al.'s attempt to make things rigorou.

As to your second and third examples, I don't think they are examples of people spending much time in places that their successors labeled blind alleys.

Sure, aspects of their work ended up being useful, but other times books and papers had results that relied on "theorems" that simply weren't true. For example, many took for granted that a continuous function had to be differentiable almost everywhere until Weierstrauss gave counterexamples.