Whenever the theory of "leading" applies, the chord should be named according to the leading note, as elaborated above. The spelling of the accidentals will then reflect towering thirds: Cdim7 would be C-Eb-Gb-Bbb, correctly leading to Db major.

However, these chords have many other functions. The progression Cdim7 → C is very common, and the natural spelling of that Cdim7 would be C-D#-F#-A. To make this into towering thirds, we must start with D#, so that the "correct" name would be D#dim7, but it does not lead to E major, not even "implied", "elided", "subsidiary", or "replaced".

In such situations, I find the chord is best named according to its bass note. Chord symbols are very rough approximations of the actual harmonic content, only good for practical purposes, even in cases like Kumbayah.