The original question mentions beginning students who are defeated by such statements as $\ker(AB) \supseteq \ker(B)$, and I suppose that it is this student population that the original poster ACL is concerned with, rather than more advanced students.

Such students need a very little bit of set language (not axiomatic set theory). And they need a very little bit of formal logic: logical connectives and quantified sentences. Maybe 10 pages altogether of what used to be chapter 0 in every more or less advanced math text. And they ought to be shown how to translate between set operations and logical connectives, e.g. AND $\leftrightarrow \cap$.

(Actually, it seems to me that I had an entire career in mathematics without really needing much more about foundations than what I just mentioned, and Zorn's lemma.)

I think the students to whom ACL refers almost surely will not know the definition of linear independence when they finish their linear algebra course because they do not understand how to use logical connectives and quantifiers.

It baffles me we don't teach these things explicitly and repeatedly, early and often, since one cannot actually do any mathematics, even at the level of a first linear algebra course, without this much "grammar", as Ronnie Brown calls it.