Then the entire world press copies the article within hours. Even Reuters and AP have adopted the practice.
Then Wikipedia cites the narrative as the truth.
Hold on, when is it you think that newspapers did _not_ do this?
In general, if a paper is writing about anything vaguely contentious, it will use unnamed sources; if it names its sources then its sources won't stay sources for long, and the media will become little more than a system for regurgitating press releases. It is always worked this way; this isn't new.
(There was a fun bit in "Yes Minister" where the minister, while leaking something, was offended that the journalist wanted to use "sources" instead of "sources close to the Prime Minister" to attribute his leak...)
[0] and submarines, of course ;)
Just like journalism is full of individuals paid off or paid to say (or not) something (sometimes blantantly untrue), wikipedia has plenty of inaccurate or false information.
So, it used to be I would use wikipedia as a type of third hand source - find information on a topic, then dig through it's references, then list Wikipedia as a source that I had used, but never quote Wikipedia.
The problem being there are few encyclopedic sources on the internet to begin with, so when the articles start sourcing the article aggregator, then the quality is bound to go down the tubes as well.
Technically, Wikipedia is a tertiary source. A "third hand source", as you say.
A secondary source is something like e.g. a book written by a historian based on research they did translating and putting together some primary-source ancient writings. The secondary source can be considered semi-authoritative on a topic; but is still interpreted through a lens. You can quote a secondary source, though always with attribution. What you cannot do, is to state claims from a secondary source as [cited] fact (like you can with a primary source.)
Because of the "anyone can edit" part, though, Wikipedia has no authoritative-ness to it — nobody is standing behind and vouching for the validity of any given text that exists on Wikipedia; there's nobody to take responsibility for the inaccuracy of a statement, nobody's professional scholarly or journalistic or critical reputation is on the line. So it's not valid to quote Wikipedia even as a secondary-source "attributed fact." So it's not a valid secondary source. Thus, tertiary.
Mind you, it's also bad even for a tertiary source. A dictionary is a tertiary source, but writers on grammar like https://www.grammarphobia.com/ might still cite historical editions of dictionaries to prove extant historical understanding of a meaning of a term. Not to take the dictionary as authoritative, but just to, effectively, "do cultural anthropology to it" — seeing the dictionary as a well-known work of writing at the time, that can be analyzed for its word choices regardless of who wrote it. In theory, you could cite [a particular point-in-time snapshot of a page from] Wikipedia for similar reasons; but it's one of those rare exceptions where you have to really know what you're doing. You might say that it's challenging to cite Wikipedia, both in a technical sense, and in the sense of doing so being the best thing to do.
Giving indications about the source is important for the reader to evaluate its seriousness. Not giving too much information is important to keep your source long term, and to get others (nobody is going to talk to a journalist know to expose their sources). Of course this depends on journalists being reasonably truthful otherwise the whole thing has no value. This is why reputation is critical and serious journals sack their journalists when they find out they lied.
What is the alternative? It is also very easy to make up a quote, attributing it to someone who’s never said anything like it or just make up a name.