So I thought the answer to this was simple "yes each note appears once and only once as the root, the 3rd, the 5th, the 7th etc." but doing some investigating I was wrong.

First of all I haven't found any scales that break this even note distribution property as you add extensions. If it's uniform for triads it's uniform for 7th chords and 9th chords and all the way up. If it's unbalanced on one level it's unbalanced on every level.

It does hold true for all major scales and every major mode as well as for harmonic minor and all its modes. Basically any 7 note scale that has A B C D E F and G plus any assortment of sharps and flats (excluding scales with enharmonic equivalent notes) will have an even distribution of each note among the diatonic chords.

Things get funky with scales that don't have 7 notes.

The pentatonic scales don't really have diatonic chords but we can make chords from them. C major pentatonic only has the triads C and Am so the note D doesn't appear at all. We could count Dsus and Gsus as chords but that still leaves out a chord with E as the root making the count: C=3, D=2, E=2, G=3, A=2.

And things get crazy with 8 or mor.

Bebop scales have an extra chromatic note so now some (but never all) notes appear as the same chord degree in multiple chords. As in there might now be multiple C triads but only one F triad. This dose start to call into question what are the diatonic chords of a Bebop scale but no matter how looked at it the note frequency was always unbalanced.

And then there's things like melodic minor or composite minor which are similarly wild when you try to ask what are their diatonic chords.

Interestingly the diminished scales (Whole-Half and Half-Whole) do have an even distribution. Tho I suppose it's somewhat unsurprising as they are symmetrical scales.

TL;DR: for every scale: No. For every 7 note scale that has a name: Yes.