In tonal harmony:
the Dorian mode cannot be a tonal center because this quality belongs to the relative minor of the Ionian scale.
the Lydian mode cannot be a tonal center because it possesses only one tonic.
the Locrian mode cannot be a tonal center because it does not possess a proper V.
The piece K1 uses the C# Dorian mode, not C# minor; in modal harmony, saying C# minor means nothing. In tonal harmony, saying C# minor means the relative minor of E major. The A# to which you refer is a structural part of the C# Dorian mode.
The C Ionian scale is called "diatonic" because it has two tonics, C and F. All scales are diatonic except for the Lydian.
All music is the expression of a state of mind. In particular, modes are attributed a state of mind; it is clear that interpretation can change based on the musician's sensitivity.
If the cadential tendency is towards the dominant triad and not towards the tonic triad, the importance of the tonic triad as a tonal center is lost. This is modal harmony: the elimination of the roles that modal scales have in tonal harmony.
If we derive modes from a Lydian scale instead of an Ionian scale, the relationships between the modal scales and the tonic triad change.
The relative minor is always on the VI degree of a major scale. In the Ionian system, for example, the relative minor is on the VI degree and is an Aeolian scale. In the Lydian system, the relative minor is obviously always on the VI degree but it is a Dorian scale. This is why in the Ionian system the Aeolian mode can be considered a tonal center and Dorian cannot. Conversely, in the Lydian system, Dorian is a tonal center and Aeolian is not.
Anyway, I realize that we are trying to climb a scale starting from the ......10th step.....
Be careful, I do not want the application of the Lydian concept to tonal harmony to be confused with true "modal harmony."