It's a tendentious question, certainly. It might mean, if Bourbaki, let us say, had had more of an interest in lattice theory, that the French word for "lattice" of this kind would be more familiar at least to me (it's treillis, which is not overloaded in the same way that the English word lattice is in mathematics).
Bourbaki wasn't very interested in logic, and hardly interested at all in combinatorics. Which is why Rota's comment is meant to sting a bit. Let's just assume, to keep a cool head, that the "Bourbaki" viewpoint from around 1965 is something of historical interest, not a description of how mathematics is or should be 45 years later. (It's an obvious remark, but needs making. We'll get back to Grothendieck in a moment, but I choose 1965 as a date at which the equation Grothendieck=Bourbakiste ultra would be at its most plausible. He was both more and less than that. Less because he came to regard writing up foundational material as a chore.)
Bourbaki claimed descent from Hilbert, but "reception theory" applied to French importation of ideas (particularly German thought, but this goes back to Newton and Locke) is never that simple. For logic, there was some joy in pushing aside transfinite induction, with Zorn's lemma, because ordinals could then be dropped. The negative part of Hilbert's legacy (lack of conscience about non-constructive arguments) was accepted, the interest in metamathematics not.
So we come down to a few questions:
1) To what extent is the lack of interest in logical infrastructure in this Bourbaki-1965 outlook a matter of sheer prejudice and arbitrary decision-making probably to be traced to early Bourbaki from the late 1930s and so to Weil?
2) To what extent, on the other hand, is it a principled approach to the overall conception that mathematics belongs in general theories, axiomatized in say ZFC, that this is Cantor's Paradise as spoken of by Hilbert, and logic fortifies Eden rather than inhabits it?
3) Is the expansion of general theories, such as is seen in SGA in an extreme form (some would say, not all though), really contingent?
The last one seems the good question for historians. To some extent the agenda for mathematics is laid down by "tradition", to some extent mathematicians update it by editing the list of traditional problems (say odd perfect numbers out, Weil conjectures in), and to some extent applications drive change. That assumes the agenda is phrased as concrete problems, not "we'd like a theory of X". But perhaps we would. We can take mathematicians' talents and motivations to be possible contingent factors; but we can't assume that those talents are so portable as to be as good for area B as area A.
By the way, the answer to Q3 seems clearly to be "some truth in this".
In the end, could Wagner have written symphonies that became part of the repertoire, as most of his operas have? How counterfactual would we have to be with his biography to make that more plausible? It wasn't that he needed to know more musical theory. Rota is wanting "alternative history" for mathematics, but like many who play with alternative history, he's making some kind of political point.