The circular ('Circle of Fifths") arrangement shows the true relationship of keys and chords to music in general.

It is only true that D flat is enharmonic to C sharp if you are playing stuff like Schoenberg's First Chamber Symphony (which does actually require that). The circular arrangement is making an assertion about tonal relationships that only applies to certain types of music.

Does traditional music modulate keys at all, Jack?

Just in the Shetland repertoire, "Da Slockit Licht", "St Anne's Reel" and "Miss Susan Cooper".

In the Scottish danceband repertoire there are a few dozen commonly played tunes that use the trick of modulating to the dominant and subdominant for whole sections - "The Bluebell Polka" includes an extra trick with a brief move to the dominant of the dominant.