Surveying the internet content on diatonic accordions, I have encountered the frequent refrain that 3 row button accordions give the player extra versatility -- at a price. Due to the greater ease with which they play chromatic passages, and because of the availability of notes in both bellows directions, they sacrifice the "charm" of the simpler one- and two-row diatonic boxes. The unsuspecting player, so the argument goes, will find his playing "smoothed out" and will soon gravitate toward music that should never have been played on a melodeon.
And then there is the reality. Many of these boxes aren't all that "big." The distinctive effect of rapid bellows changes is still available. The bisonoric basses add harmonic interest. And even the biggest of these expanded-range boxes are still centered on "home" keys and related minor keys in a way that, like a one-row accordion, gives different weight to each note.
If you don't believe it, ask Cyril Roche. (Video from YouTube.)