Although the convention does go back much farther than the 1940s, I think you’ll both find that, far from dying out, it is still the accepted standard in most major style guides. In a quick Google, I found that AP, APA, Chicago, MLA, and NYT all support it when block quoting is not possible or desirable.
Although I think I stated the reasons for the convention fairly clearly above, here is another explanation from Stack Exchange:
“That seems like an odd way to use punctuation,” Tom said. “What harm would there be in using quotation marks at the end of every paragraph?”
“Oh, that’s not all that complicated,” J.R. answered. “If you closed quotes at the end of every paragraph, then you would need to reidentify the speaker with every subsequent paragraph.
“Say a narrative was describing two or three people engaged in a lengthy conversation. If you closed the quotation marks in the previous paragraph, then a reader wouldn’t be able to easily tell if the previous speaker was extending his point, or if someone else in the room had picked up the conversation. By leaving the previous paragraph’s quote unclosed, the reader knows that the previous speaker is still the one talking.”
“Oh, that makes sense. Thanks!”
This example is set in the context of dialog, but the same principles of clarity and readability apply when writing non-fiction as well.
In the context of computer coding, open quote marks or brackets or parens without corresponding closing marks generally lead to problems, and so in that sense the convention can be seen as “illogical.” But readers are humans, not computers, and readability tops rigid “logic.”
If the convention is dying out (and I sincerely hope it is not), it is only because so many people do not know or understand it and use the “wrong” version in settings like this. Hence my nitpick. Just trying to fight ignorance.
But if it really does bother you, the solution, as I said, is to use block quotes wherever feasible.