I agree with most of that Jerry but "English dance tunes often are very chordal in nature"?...Morris, e.g., was originally accompanied with just pipe (tune) and tabor drum (rhythm)...yes?
Sedayne - I do look forward to a singaround with you and your stylophone.
To Captain Swing - never been into train spotting, although they are my favourite form of transport/travel...in a coach one can barely swing a...
And I was going to mention trad Indian music, John, which has generally/always? been one note at a time, yes?...and that's why many there have taken to the hand-pumped version of the French harmonium, yes? Also, just the other day on BBC Gaelic radio e.g., I have enjoyed the tune being intoduced on bagpipes, followed by singing over the drone, as you suggest.
I've never tried, but I'd guess electrical keyboards can be set-up similar to the way you suggest for harmoniums - one key plays that note plus a fifth or an octave above it, as well, yes?...that would give a thicker and, some would say, richer/warmer sound but, again, I'm afraid, I'd probably rather just sing with the top-line melody only.