The full score can be either transposed or in concert pitch. This comes down, as with everything, to conventions set by common practice. Scores up to the twentieth century are almost exclusively transposed. I’m sure there are exceptions, but, off the top of my head, I don’t think anything beyond a composer’s manuscript or a particello was notated in C, and even those were probably written in transposed pitch already. Brass, for example, kept being written without a key signature even when valved instruments were common, in the 18th and 19th centuries. Eventually, as common-pratice tonality expanded, brass (and timpani) started getting key signatures, but stayed notated in the score as it appeared in the parts.
Only in the advent of the 20th century did scores in C become anything close to convention, and even then, I believe it grew out of the Second Viennese School. I can’t assert with all certainty that there’s all there is to it, but I can tell you that the practice was paradigmatic in that tradition. Scores in C almost always have the caveat that octave transposing instrument are in their written, and not sounding, pitch.
A conductor has always been expected to be in firm command of these conventions and to be able to calculate transpositions on the fly. Composers were as well, hence me saying before that a composer’s manuscript would be most likely transposed already if it was anything beyond a sketch or particello. No matter what the type of score, the conductor is expected to address, say, a horn player using his written pitch, even reading off a score in C. Besides, the way the score is notated is a choice made firmly in the camp of the composer. I can hardly imagine a conductor asking the composer to switch things around.