Standard disclaimers apply here (all my opinions).
This is actually a huge topic, one at the heart of efforts to save societies from parasitic and malevolent falsehoods. If we really wanted to tackle it systematically at PS, we’d need some structure. I don’t think that’s realistic right now.
In this particular case, the posting of an advertorial as though it were a news article, we have a reasonably straightforward case study. The piece is clearly marked as an advertisement, and is written as an advertisement. There is a transition point when the advertiser talks about their product (in this case, they recommend products, which happen to include the product used in the story).
Is it appropriate to post a link to an advertorial or other advertisement, and then to ask questions and invite discussion? Sure, but with that made explicit. “Here is an advertorial at USA Today. They seem to claim that some people may have arrived in the Americas by a different route than previously thought. Is that true? Or interesting?”
Personally I would prefer we do that rarely. In this case, further examination reveals that there’s really no science in the story, that even the meager press coverage is not interesting and emphasizes the testing company, and that there was no followup by scientists after more than a year. All of that could have (and did) come out of a discussion of the advertorial, but ideally we would never have had to discuss it since the site was clearly marked as something other than news.
I’m not sure how much more to add because I think we all have to live with errors like this. The best approach IMO is for the poster to be able to say “Oh I see that it’s an advertorial, sorry about that, I’ll make that clear in the title” or something like that. What we should not do, IMO, is ignore the error. And we didn’t.