Just wondering if you ever got this figured out.  I'm in an identical situation with similar Murray equipment (early 90'ish Murray - 38 in/11hp briggs strat).  And the solenoid only clicks if I let it sit for an hour or so.  Can't jump from car battery or from portable battery.  I can get the starter to spin for a few seconds if I jump straight to starter with portable battery.  Didn't do near the level of testing you did, but I know I have 12 volts from the battery to the starter with the ignition in the start position and all the safeties bypassed.  Fuse looks good and passes continuity test.  I also have 12 volts going into and out of the solenoid on the main positive posts (in from battery and out to starter).  I haven't put the dvm on the ignition yet because ............ well ............ it's kinda difficult to get to 

  ..... but I guess I'll go there next.  Also haven't tried shorting the solenoid yet.  There's a little bit of rust and corrosion on some of the connections but nothing that looks too bad.