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.
When are you claiming all this work on 1-1+1... ended? in the 17th century as in your original comment, or with Cauchy? Do you really dispute that Gregory in the 17th century claimed that not all series should be summable?
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.
You seem to be saying that sloppy proofs of true theorems are not useful.
So apparently Richard Feynman once said:
I could be missing something, but this strikes me as a terrible answer.
When was the atomic hypothesis confirmed? If I recall correctly, it was only when chemists started noticing that the outputs of chemical reactions tended to factorize a certain way, which is to say that it took millennia after Democritus to get the point where the atomic hypothesis started making clearly relevant experimental predictions.
How about, "Stop trying to sound wise and come up with theories that make precise predictions about things you can measure in numbers."
I noticed this on Marginal Revolution, so I shall also state my candidate for the one most important sentence about macroeconomics: "You can't eat gold, so figure out how the heck money is relevant to making countries actually produce more or less food." This is a pretty large advance on how kings used to think before economics. I mean, Scott Sumner is usually pretty savvy (so is Richard Feynman btw) but his instruction to try to understand money is likely to fall on deaf ears, if it's just that one sentence. Think about money? Everyone wants more money! Yay, money! Let's build more gold mines! And "In the short run, governments are not households"? Really, Prof. Cowen, that's what you'd pass on to the next generation as they climb up from the radioactive soil?
*Cough.* Okay, I'm done. Does anyone want to take their own shot at doing better than Feynman did for their own discipline?