We have in an 1858 court case:
"A curious doctrine this,—a singular kind of subtraction,—to subtract crime from crime, and there remains nothing but innocence."
It appears that we are no longer "allowed" to do this. But why not?
If you go back another 150 years, you'll see passages like this one from Lives, English and Forein : Containing the History of the Most Illustrious Persons of Our Own and Other Nations, from the Year 1550, to the Year 1690 (1704):
The book consistently separates sentences by a quad space, and you can observe certain other admirable consistencies:
Why aren't we allowed to follow these rules today?
Objectively, of course, we can follow any rules we want, or no rules at all. But practically speaking, we follow contemporary conventions, just as the judge ordinary in Hope v. Hope did in 1858, and just as the "several hands" responsible for Lives, English and Forein did in 1704.
The simple explanation for why punctuation like ",—" fell out of favor is that writers noticed that they didn't need to use both a comma and a dash to indicate the kind of break they had in mind.
As for why anyone ever used ",—" in the first place, I speculate that the comma came first in accepted punctuation; and then, when someone wanted to express a more radical break in the flow of ideas than commas normally indicated, he (or she) chose to add a pair of dashes to the existing (and already standard) commas. Only after the novelty of using dashes in this way wore off would the question arise, why keep the commas at all? But this part of my answer is pure speculation.
In a comment, the poster equated using ",—" instead of either a comma or a dash alone with spelling weird with the e and the i reversed. I think, to the contrary, that using ",—" is more like using the spelling weyard (as Shakespeare did) in place of the modern spelling weird: I wouldn't take it as a sign of ignorance or poor education (depending on the context in which it appeared), but I would see it as something of an affectation—an intentional archaism.
There is something called "overpunctuation". I once heard a professor of English refer to the 2nd Amendment to the US Constitution as "overpunctuated".
As passed by the Congress and preserved in the National Archives:
As ratified by the States and authenticated by Thomas Jefferson, then-Secretary of State:
Both versions mean the same thing, but the first contains unnecessary commas -- there is nothing "wrong" with the commas, but they are unnecessary. This is likewise the case in the passage you reference in your question. The dashes suffice to separate the subpassage and the additional commas are superfluous.
We should probably avoid superfluous punctuation in our writing.
Writing has changed considerably since 1858. Today we would write it without the commas.