I think that the case for the rule against separating independent clauses with a comma in the "According to John Smith, independent clause A and independent clause B" instance is far stronger than the case for a comparable rule in the "Before the Humane Society opened:" instance.
My reasoning is that the source attribution situation is far more likely to be truly ambiguous, since in every sentence of that form the named source might be responsible for the assertions in both independent clause A and independent clause B, or only for the assertion in independent clause A (with independent clause B constituting an observation by the author that the named source did not say or endorse). For example, in the sentence
The President said the Premier's wife is a very tough negotiator and everyone knows what a liar she is.
the comma (or lack thereof) is rather important to helping readers determine with confidence whether the President's remarks ended at "negotiator" (in which case a comma is appropriate there) or at "what a liar she is" (in which case there should be no comma after "negotiator."
But the sentence
Before the Humane Society opened, the cats in the city were in a very bad spot and the dogs that lived around the county were arguably in a worse one.
doesn't present a similar basis for misunderstanding, whether you include a comma after "spot" or not. The biggest reason for the difference is that in the "Before the Humane Society opened" sentence, the introductory clause doesn't set up an attribution that might expire without warning after the first independent clause. Rather, it sets up a time frame that will remain in force until the author explicitly identifies a new time frame. You don't need to avoid separating the two independent clauses with a comma, for the same reason that you don't need to reinforce the "Before the Humane Society opened" time frame in every subsequent sentence (and there might be paragraphs or even pages of them in some cases) where that time frame remains in force.
I'm sure that it's possible come up with instances where the "Before the Humane Society Opened" introductory phrase was intended to apply to only the first of the two independent clauses that follow it. But those instances are better handled by marking the shift in time frame for the second independent clause rather than leaving the comma to do the job. The strict rules governing comma use in the source attribution situation are better seen as being designed to handle a special case than as a model for all sentences that have an introductory phrase and two independent clauses.
To sum up, I suspect that you haven't found a comma rule comparable to the one involving source-attribution introductory phrase + two independent clauses for the more general case of any other type of introductory phrase + two independent clauses because no such rule exists for that case. And it doesn't exist because it isn't really needed except in the special case of source attribution.