Frege disagrees:
"Just as the concept point belongs to geometry, so logic, too, has its own concepts and relations; and it is only in virtue of this that it can have a content. Toward what is thus proper to it, its relation is not at all formal."
I think this is right, but Hilbert came already close much earlier.
But "Before this, logic had been strictly syntactical and proof theoretic" is at least misleading.
Traditional logic was more conceptional. Frege's "formalism" was a great innovation, but he remained within the conceptual tradition (Frege-Hilbert controversy).
"What Hilbert offers us, in 1899, is a systematic and powerful technique that can be used across all formalized disciplines to do just this: to prove consistency and independence. In doing so, he lays the groundwork, in concert with various of his contemporaries, for the emergence of contemporary model-theoretic techniques."