Gurney: yes, that flatness can certainly happen with a treble concertina. A metal-ended one usually has more of a cutting edge, but that's not what you want for song.
For what it's worth, I'm a sort of weedy alto and manage to sing audibly with a combination of treble English, McCann duet and baritone English with Spare Parts. I do have to keep asking the other two to keep the noise down, though.
I'd agree that the tenor English would be the best choice for someone accompanying themselves.
Which brings in another point: a good English concertina can play softly as well as loudly. It's a myth that concertinas only do mezzo-forte.