somewhat of an update. I had some more time to check things out. What I'm seeing on my MPC is that the Sync Out signal (on pin 10 of the connector of the sync jack board) is essentially being dumped into the ground plane.
So, if I unscrew the sync jack board from the MPC, the problem goes away. Measuring the voltage on the sync board mounting bracket versus the MPC chassis shows +4.6v DC. On the scope I see a rough triangle wave of about 2v pp AC.
If you look at the schematic for the jack board (labelled page 22, it's the 36th page in the PDF file), you can see Akai has 3 grounds on that board. Regular chassis ground, Analog ground, and a 3rd one labelled FG (floating ground?). From what I see, it looks like the Sync signal (which is a 4v square wave) is being filtered by the resistors and cap on the way to the sync out jack and is showing up on FG as the rough triangle waveform. In the bottom right corner of that schematic, you can see where FG, AG, and GND are all tied together though a pair of 0.1uF caps. That dirty ground is making its way onto that metal bracket and when it is screwed into the MPC, causing the problem.
Since this scheme used to work, I'm still trying to figure out what has failed/changed to cause this issue. The caps seem to measure OK, but I didn't take them out of circuit.
As a perk for reading this far, http://www.oldgearguy.com
has a very useful link for those that are techically inclined with regard to the MPC 3000.