OK, perhaps I'm overstating the case, and building something of a strawman out of the extreme position. However, even if most practitioners attempt to mix the two approaches, they've still missed the point of the observation.
It's called the "Principle of Compositionality" and it's summed up excellently in O'Reilly's book by Steven Bird, Ewan Klein, and Edward Loper on the Natural Language Toolkit for Python programming:
the meaning of a complex expression is composed from the meaning of its parts and their mode of combinationThere's a deeper examination of the term at Wikipedia, but Bird et al's summary is pretty clear and correct.
So a language isn't just a collection of words. It's a collection of words and a collection of ways of combining words. (Ignoring the fact that a "word" is often a combination of smaller morphemes.)
Teaching individual phrases as fixed units leaves behind much of the subtle, beautiful complexity of how languages build up their meaning.
In English teaching, it is often claimed that so-called "phrasal verbs" are not systematic and must be memorised, but what we do with "verb + particle", the Romance languages do with "prefix + verb root". A fire extinguisher puts out fires, and we shout out our exclamations. Seems pretty systematic to me. (Not to mention German, where a prefix often becomes detached from its verb and becomes a particle -- see? it's all part of a single spectrum....)
And when people talk about the arbitrarity of "to be" vs "to have" in ages (en "I am 33" vs fr, it, es etc "I have 33 years"), well, at least it's consistent within the language. It's a logical consequence of the Romance "to have" structure that phrases like "at 40" (life begins...!) become "with 40" in these languages.
But while most learners are capable of getting a handle on the be/have difference, I still meet a great many people who borrow the "with" structure into English. How easy would it be for the teacher to point out a few of these little things? To encourage the learner to build a meaningful model that (at least in part) mirrors the native speaker's one?
But perhaps that would take too much time. Nevertheless, we have built an environment where we discourage our students from looking for meaning and structure. We expect them to resign themselves to learning everything as an arbitrary single data-point.
That subtle, beautiful complexity I was talking about? We hide it from them. We keep it from them. We make learning a language into an ugly, clumsy drudge. What we are hiding from them isn't just thte beauty of the language, because that beauty is intrinsic to the language. To hide the beauty, we must hide the language.
How can we teach someone a language we are unwilling to truly share with them?