Well, some modernists did take objectivity to an extreme. Hegel for one, whose absolute rationalism sparked the existential backlash in Kierkegaard. Richard Rorty showed how modern, naturalistic philosophers and scientists often implicitly assumed the mind is a "mirror" of nature. This showed up in the correspondence theory of truth. The mental or linguistic representations "agree with" the facts.

In any case, the question would be the nature of even imperfect objectivity. Does objectivity require detachment from all contexts, the availability of a transhistorical or impersonal perspective?

I think objectivity is possible, and the "postmodern" (hypermodern) criticisms clarify how objectivity might work.