I think it can help, as long as your playing is at a level to merit the supplement. You might want to cue the best three minutes. I know Harvard appreciated that (we asked them).
In your case, if other parts of your application and/or resume don’t reflect your level of playing (no awards, no impressive ensembles, etc.) then inclusion of a recording might help.
A supplement might include the recording plus a music resume, repertoire, awards, programs or articles, and perhaps most importantly, letter (s) from music teachers and/or directors who can not only attest to your performance ability but aspects of your character such as work ethic, cooperation, love of music, intellectual interests etc.
Harvard has a double degree program with NEC, which has a jazz program. If a double degree appeals at all, check out Oberlin too. Remember that Yale’s School of Music is a graduate program 
Columbia sounds like a great choice for you as long as you have some safer ones too.