I was having another look at Gerhard Antretter's Deutsch zu Zweit, and decided to see what he had to say about articles. The index gave only a single page for the subject, so I feared the worst.
Was die Artikel ein und der betrifft, so stellt für bestimmte Lerner nicht deren Deklination das Hauptproblem dar, sondern die Tatsache, dass es sie überhaupt gibt. Neben sehr vielen anderen Sprachen kommt beispielsweise das Russische ohne solche Wörter aus und erfahrungsgemäß ist es das größte Einzelproblem russischer Lerner, sich auf die Notwendigkeit ihrer Verwendung im Deutschen zu besinnen. ... Wenn Ihr Partner zu diesen schwierigen Fällen zählt, suchen Sie sich gute Übungen zum Thema und legen Sie große Beharrlichkeit an den Tag. Wir gehen wegen der großen Komplexität der Materie und des im Verhältnis dazu doch nicht so großen Gewichts der Fehler nicht näher darauf ein. (!!!!!!!!!)
That is [rough and ready translation],
As far as the articles ein and der are concerned, it's not their declension that presents the main problem for certain learners, but the fact that they exist at all. Among many other languages Russian, for example, has no such words, and experience has shown that the single greatest problem[!!!!!!!] for Russian learners is remembering the need to use them in German. If your partner is one of these difficult cases, look for some good exercises on the subject [!] and show great perseverance [!!!!!!!]. Because of the great complexity of the material [?!] and the relative unimportance of the mistakes [hm], we shall not deal with this in greater detail. [punctuation fails me]
Now the thing is. The point of Deutsch zur Zweite was to provide a book that couples could use as a resource when one was German, the other not, so that the non-German could improve by practising with someone who was (probably) not a trained language teacher. A very large proportion of foreigners in Germany come, as it happens, from Eastern Europe; another substantial number come from Turkey; both groups have a first language in which articles are not used. And Antretter's right - for someone who is used to surviving quite happily without any articles at all, working out where to deploy them in a language which uses them with gay abandon is probably the single greatest problem they will encounter. So, um, wha-?
If your native language uses articles, TEACHING someone how to use them is probably one of the trickiest things you can tackle. It's not easy for professional language teachers; for an amateur it's unbelievably hard. So hard, in fact, that it's precisely the sort of topic the amateur might hope to find covered in, um, a book about teaching your partner to learn German. (Telling the reader 'Well, this is incredibly complicated and we can't be bothered so we'll just leave you to your own devices' is, to put it mildly, unhelpful.)
Is it really the case, anyway, that these mistakes don't matter much? In English, in popular imagination, the linguistic development of mankind matches that of the English speaker. Once, in the dawn of time, hominids roamed the earth, using primitive implements of flint. They had primitive terms with which to identify objects in their environment, yes: stone, knife, woolly mammoth. But not only did they have no notion of modern dentistry and supersonic flight, they also lacked the technology for definite and indefinite descriptions. The definite and indefinite article are the products of thousands and thousands and thousands of years of linguistic evolution... And in our infancy we retrace the linguistic development of the race. We too start with simple nouns and adjectives, the odd verb, and work our way up to articles, adverbs, tenses and other jollities. The non-native speaker who dispenses with articles sounds like someone at the stage of linguistic development of (at a guess) a two-year-old. Or Australopithecus.
Adam Smith, I think, said the basic requirements for human life include what is necessary to participate in society without shame. If you live in a society where all men wear hats, a man who does not have a hat lacks a necessity - he cannot appear in public without shame. Similarly, if failure to use articles slashes your apparent intellectual age to that of a toddler, the use of articles is a necessity; a book for couples confronting an article-heavy language ought to tackle them.
(It may be, of course, that the omission of articles in German does not have the Australopithecus effect. I've noticed, though, that Germans have a pronounced antipathy to teaching their system of articles to outsiders - it's always either jam yesterday or jam tomorrow, something either so simple you should have covered it in a beginners' course or so complicated it must be gone into at great length some other time.)
DzZ does have a good section on separable verbs, also normally jam yesterday/tomorrow.