Pip,
It's perfectly easy to make a reasonable song accompaniment with that easiest of instruments, the melodeon. Tony Hall has done this most of his life and I've never heard any complaints about him. It is very true as earlier stated that it is 'difficult to play it really badly or really well'. And there are melodeons in DGA. But as others have said if you want to play chords and drones then a cheap 48 button wooden ended quiet English concertina is perfect, and as Button said, button for button, or reed for reed, English concertinas are much cheaper than equivalent model Anglos simply because of supply and demand.

As a middle of the road player/singer I accompany my singing on both Anglo and melodeon and the different sounds they produce make for variety. But, as with any instrument you need to get the feel of the instrument, i.e., get to the stage where your fingers are playing the tune and not your conscious mind, before accompanying song successfully.