So to answer the question “Why play DADGAD at all” here’s an explanation.
With DADGAD, in general, the chords have the bottom string ringing out the root of the chord. Hard to get that same sound, for example on a D chord (the most common chord in Irish music) with EADGBE. Take a listen to what John Doyle does on Solas albums. He’s using drop D generally is my understanding (DADGBE) and uses the damped D string to give the music that pulse, like a bodhran tuned to D. Drop D has its advantages as well for combined backup and melody playing.
Lets take a look at some of the chord shapes I use, for example for D Major backup:
D Major -
Fretted: 050020
Notes: DDDAAD
So that’s a power D chord with no third. How handy, can use that for both major and minor, sound great.
For the G chord (IV):
Fretted: 550000
Notes: GDDGAD
So that’s a G chord with no third and a high A (giving a 9th suspension that sounds great). Lots of D strings reinforcing the D tonal root of the tune. You can also fret the A string at 2 if you want a full G chord, but I like the sound of the A in the chord, reinforces the D drones.
For the A chord (V):
Fretted: 777000
Notes: AEAGAD
Wow, all kinds of stuff going on here. Its an A chord, again, no third, with a G giving a 7th sound and the high D drone keeping some continuity of the tonal center on D. If you want a full A chord, you can fret the G string at 6, giving a C#, but that tends to be dissonant against the high D, so I leave it out.
If you write out the rest of the shapes, you’ll see that have similar attributes. They tend to leave out the third, and generally have either 7 or 9s added for color and coolness.
Of course, with the capo, you can put these same sounds in any key. Notice the nearly complete lack of thirds in the chords and the droning strings reinforcing the tonal center of the tune.
Here’s the same thing for a dorian pattern.
D minor :
Fretted: 050020
Notes: DDDAAD
Looks familiar, no?
C major:
Fretted: 035500
Notes: DCGCAD
That’s a C chord, no third, droning against an open 5th D chord.
B-flat major:
Fretted: 01 33 00
Notes: DBbFBbAD
or
Fretted: 8 88 700
Notes: BbFBbDAD
Both are Bb major chords sounding against a droning D power chord with no third. How convenient that the D is also the third of the Bb chord.
Again, notice that there is always a D drone throughout all the patterns, reinforcing the pedal tone of the tunes.
One could do the same thing with standard tuning with more complicated chord shapes and fingerings, but DADGAD make this really easy and sounds very appropriate for modal scale based Irish music.
Hope that makes things a bit more clear… 
Cheers,
Michael