A diminished 7th chord is built with two tri-tones, which means it could represent four dominant chords:
Let's try to fit these chords to the key of B-flat, which is the key of the moment of this song, Someday My Prince Will Come. This is not a theoretical analysis since we are not considering what chord might follow.
Which one of them has all the chord tones of C#dim7? The answer is A7(b9,b13).
Let's compare the scale:
If you add C natural, which is a #9th of A7, and often added when the b9th is used, the resulting scale is identical. Coming up with a cool phrase on A7(b9,b13) is a lot easier than sight reading C#dim7 and trying to figure out the non chord tones according to the key of the moment.