Arnold Shoenberg, in his Theory of Harmony, shows that the major diatonic scale is constituted by the major triads of the tonic, dominant and sub-dominant tones. For example, this means that the C major diatonic scale can be constructed with just the tones in the C major, G major, and F major chords.
He points out that the notes of a major triad are exactly the notes of the first three unique harmonics, indicating a possible source of the power of major triads. It has also been recognized as early as Pythagoras that tones whose frequencies lie approximately at small integer ratios sound "consonant" and pleasing when played simultaneously or soon after one another.
As to why such a revered scale should be based around the tonic, dominant, and sub-dominant, we can again take note of the harmonic series and Pythagoras. The dominant sits at a frequency ratio of 3/2 with respect to the tonic, taking after the second unique harmonic in the harmonic series based on the tonic. Relatedly, the tonic lies at a frequency ratio of 3/2 to the sub-dominant.
If you have access to a standard piano keyboard, try playing just the C (tonic), F (sub-dominant), and G (dominant) tones in various sequences and rhythms. Try comparing the affect of the tonic alongside various other pairs of tones (C, A and C#, for example). I've found that no other combination of just three tones better serves to instil the feeling of "home" in the note C.
Also, play the C major scale, one note at a time, up and down in a variety of rhythms. Since the major scale consists entirely of notes from the tonic, dominant, and sub-dominant major triads, try modifying the activity as follows: alongside each note of the scale, play a chord from either the C, F, or G major chords, choosing only a chord that possesses the note being played at the time. So you can play C with CEG, D with GBD, E with CEG, F with FAC, G with CEG, A with FAC, B with GBD, and finally C with CEG, for example. I find that playing alongside the chords feels like a very filled-in version of playing the lone scale. Try substituting the tonic, sub-dominant, and dominant major chords with other chords in the C major scale (or even chords from the chromatic scale), still only playing a chord when it possesses the tone currently being played in the scale. It seems like playing with the tonic, sub-dominant and dominant major triads form the most agreeable and representative expansion of the lone major scale than any other three chords (even though other combinations are nice and interesting).
In light of this, playing the diatonic scale in step-wise motion seems to have an effect of alternating between the "three areas" (tonic, dominant, and sub-dominant) of a key, providing harmonic contour of fundamental importance somewhat automatically.
As a brief indication of another important factor, note the inherent constitution of minor chords and their potential activity in the major scale. The second and third tones of the major scale can be viewed as the sub-dominant and dominant, respectively, of the sixth tone. Try the exercises again, but using the sixth tone as your tonic and minor chords in place of the major tonic, dominant, and sub-dominant chords (so playing the minor scale with respect to the sixth tone in our original major scale).
Those are some points that stand out in my mind as to how the major diatonic scale might be so important in music.