They may be called "diatonic accordions" in the US (and France), or "button accordions" in Ireland, and it's interesting to have this clarified, but the original poster made it clear that he was talking about English music and dance. In this context the instrument (most commonly a two-row) called a "melodeon", which is a useful way of distinguishing it from other accordions without having to use two words. That's what it's called here, whether used for morris or for social dance as part of a band.

A melodeon has undeniable advantages as a dance instrument. It's got the capacity for melody, bass and rhythm all at once, and its soudn carreis well. Relative to the piano accordion, its compact size and punchy sound are often preferred by dancers (I'm still talking about England, not Scotland where the piano accordion, and even the piano, are far more prevalent than in England).

This doesn't necessarily make it the ideal session instrument, as there is an optimum number of melodeons, beyond which the sound tends to become mushy and messy. Many melodeon players do play another instrument so that they can vary their contribution at a session. I switch to playing recorder (not very well, alas) at times, while other melodeon players I know also play whistle, pipes or clarinet, or simply drop out of certain tunes that don't suit their playing style. It's only really at festivals that I've seen the fearsome fifteen-or-more melodeon sessions.

Marje