I know you just stated it, but no beeps at all when you push the lanyard on the DESS post? There's a few things to check, continuity when pressing the start button, and 12 volts (yellow wire) going to the starter solenoid.
Possible the lanyard is not being recognized by the MPEM or the DESS post? Previous owners are known to get lanyards mixed up when selling a ski...you bought a non running ski, so it's bottom up process of elimination.. Go with the start button continuity and 12 volts to the solenoid as first checks.. Buzzer out is a possibility, easy to check, but a ski will start and run with it not working.