"as is the case with all the basic standard-tuning chords"   There are options in Standard tuning,Examples are for an a modal dyad, long a, there is a d modal dyad, so its possible to play in standard tuning, in d using two dyads, for the gmajor chord a more modal sound can be produced by playing the g6 and omitting the 1 string, so gbdgd doubling the first and fifth of the chord, the same goes for tunes in g major,the chord is played ceggcg six string to one, so g 6 shape is used omit 1st string, long d for major and the c chord has the first and fifth doubled again, the little finger frets one string on 3rd fret.
double droppd d is also good, the ordinary c chord gives you c9.
Martin Carthy has stated that he finds his tuning CGCDGA    more flexible for changing keys key than DADGAD, he has sais it allows him to play in four of five keys CGFD and possibly Aminor