I agree wholeheartedly--in fact, it seems to me that, by it's very nature, the term "accompaniment" requires that the tune be written first--

As to making it simpler, Gulliver(and for your consideration, Angie)--I have been playing the melody in C, using what is basically an Am/added 2nd chord, but is cut down to just A B and C, played in a row on the keyboard four to the measure-It is a bit edgier than this tune is usually done, but lends a bit of depth--