i'm also very inept at creating jazz sounds (more classically oriented), but when the 'alphabet song' came on sesame street for my three year old and some famous lady (forget her name now, it's a shame) did a jazz rendition, i thought 'ok jazz is not just notes but (as you mention) expanding on those keys/notes (making any note ok). the basic skeletal pattern is always there, but hard to find with really good jazz musicians.
wouldn't she be fairly old today? it was a younger singer, but not sure who it was.