Translation:He has a large family he takes good care of.
People say that relative clauses without a pronoun are always defining. The Czech sentence does not appear to be defining; I think its meaning could equally be expressed by "…and he takes good care of it" (I am not suggesting this as an adequate translation; it does not preserve the structure). So I would prefer "…of which he takes good care."
You are right, “people say” was not a good introduction at all. Is the following a reputable source? https://dictionary.cambridge.org/grammar/british-grammar/relative-clauses/relative-clauses-defining-and-non-defining
It says under “non-defining relative clauses”: “We always use a relative pronoun.” By converse argument, when there is no relative pronoun the clause is not non-defining but defining.
Do we agree that the Czech clause “o niž se dobře stará” is non-defining, i.e. not necessary to make the sentence semantically complete? “Má velkou rodinu” makes perfect sense by itself.
So my humble opinion is that the proposed translation, “…a family he takes good care of”, does not reflect the sense of the Czech original.
P.S: Please accept my recognition for the incredible work (amount and quality) you are investing in this course.
I declined to comment on this yesterday because (1) the alternative translation that you initially proposed -- which was the subject of the discussion -- is already an accepted answer; and (2) a discussion of relative clauses in English is beyond the scope of the course.
I do, however, appreciate your posting the link, and feel that the article would be very helpful to anyone who wants to learn more about the topic.