2 chord songs
General mountain dulcimer or music discussions
Nancy, I mention "Jambalaya" above because it is a really fun song and doesn't sound like something simple or child-like.
I think minimizing the chord changes is a good way to introduce the notion of chord progressions and chording in general. You are on the right track. When I was a wee one my mother taught me to play the ukulele by showing me two one-finger chords. She sang and pointed to me when I was supposed to switch from one to the other. That exercise helped me to hear when the chord change was happening.
So I encourage you to go forth and teach a couple of two-chord songs. But I don't think you should overdo it, meaning you don't need more than three or four. Once people "get" the idea of chord changes you can move very quickly to three-chord songs, and there are a lot of those. AsButch Ross said, if you like the first folk song you learn, you'll like the other one, too. His point is simply that all the songs we play are basically the same.
The reason Bile Dem Cabbage Down is so useful as a first song for chord/melody play is that is that it involves the three main chords that lay at the center of all the music we play. Additionally, all chords can be played with one or two fingers, and in fact the melody itself is really nothing but a few chord changes. So whenever the melody note is found on the 2nd fret of the melody string, you play a D, when the melody moves to the 3rd fret you play a G, and when it moves to the 1st fret you play an A. And fancier versions of the song are basically compilations of littlelicks to move from one chord to another, meaning they are usable in all the other songs we play, too.
I guess my point is that while it makes sense to begin with two-chord songs, you shouldn't fear moving to three-chord songs very quickly.They are not much more difficult to play but offer a lot more in terms of ear training and practical applications as well.