Editors resigning because they published a low quality paper
It's decently common. I Googled for "peer review failure editor resign" and some examples are: 1, 2, 3, 4.

You ask "Why would editors choose to publish [the paper] nevertheless?" There're many possible reasons as to why editors may choose to publish a flawed paper:


Some members of the editorial board approved of the paper and still do, while others didn't (and the ones that didn't are the ones resigning). Journals typically do not get every member of the editorial board to approve a paper before it's accepted.
They made a mistake, but now that the paper is already published, they don't consider the errors critical enough to retract it (or they don't consider it fair to the author).
They realized the paper is dubious but decided to err on the side of acceptance. People who subscribe to this idea think, "referees only give opinions; the real peer review happens after a paper is published".
The journal is currently lacking papers, so they're inclining towards accepting more papers to fill their issues.
Difficulty communicating with the author, e.g. if the author isn't very responsive, then editors might choose to accept "as is" rather than wait several months for an author response (especially if the author isn't likely to make major changes to the manuscript). I also remember reading about a flawed paper that was accepted because the author was from the Soviet Union, and getting the author to correct it through the Soviet censors takes too much time.


In the case of this particular paper, the editorial board claims no knowledge of what was happening. At this point, it's speculation to try to figure out what actually happened. However, I would guess (based on my experience in publishing) that after the journal received the paper the desk editor attempted to find a member of the board to handle it, but everyone contacted either declined to handle it or simply did not respond to emails. In that case the desk editor can hardly let the paper sit there collecting dust, so (s)he did the next best thing which was to DIY, with less than ideal consequences. I've seen this happen and even done it myself quite a bit. I've never actually ended up in the news for doing this, which shows that usually DIY-ing doesn't lead to problems. But if it does, that's when we look at one another and sigh.