Can I write a comma followed by an em-dash?
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):
Sir Walter diſpairing of the Fly-Boats, and depending on the Lord Eſſex, and Lord Thomas Howard's Promiſes to aſſiſt him, laid out a Warp by the ſide of the St. Philip to cloſe with her ; the Wind hindring him otherwiſe to Board her. When the Spaniſh Admiral, and the Commanders of the three other Capital Ships, found that the Lord Thomas, and the Rear-Admiral of the Engliſh, began to do the like : They all ſlipt Anchor, and ran a Ground, their Soldiers and Mariners tumbled into the Sea, ſome were drown'd, ſome ſtuck in the Mud ; the Admiral burnt the St. Philip, and the Captain of the St. Thomas, did the ſame by his Ship. The St. Matthew, and the St. Andrew, were ſav'd by the Engliſh boats before they took fire.
The book consistently separates sentences by a quad space, and you can observe certain other admirable consistencies:
Almost all common nouns (and three common-noun phrases—Fly-Boats, Rear-Admiral, and Capital Ships)—are initial-capped. The only common noun not so treated is side, and the only verb capitalized is Board.
All proper names are initial-capped and italicized, excluding the prefatory terms Lord and St.
All semicolons and colons have a letter space before as well as after them.
Occurrences of silent e in single-syllable past-tense verbs are replaced with apostrophes (drown'd and sav'd) or are avoided by spelling with a closed-up t (slipt and burnt).
Most occurrences of and are demarcated by adding a comma to the word preceding it and adding another comma just before the next following and or the following verb. (The only exception to this treatment is the simple noun pair "Soldiers and Mariners.")
The lowercase letter s is rendered as ſ (or in roman type ſ) when it appears elsewhere than as the final letter in a word.
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:
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.
As ratified by the States and authenticated by Thomas Jefferson, then-Secretary of State:
A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed.
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.