Solder joint issues on the KEL connector are rare, but worth checking. Cleaning the contacts may help. Visually inspecting contacts to see if any are not aligned and re-forming any misalignments is usually better then attempting to replace the connector. Replacement connectors are expensive, difficult to find, and it's a major soldering job.
The clue that DiagROM doesn't show the same issue at power on suggests that it's not related to the KEL connector. If it was the connector at fault, DiagROM would likely present the same issues at power on as the standard ROM does. I wouldn't be concerned with the fast memory not appearing with DiagROM, I see that fairly often with correctly working machines, suspect there's a few imperfections in DiagROM.
Given the clues that the A3000 is fundamentally running in the 'dead' state, and DiagROM works reliably, then most likely the system is getting hung up somewhere while booting from the standard ROM. I've seen scsi.device do exactly this when things aren't happy with the SCSI controller or some of the connected SCSI devices, possibly even bus termination.
If you have the capability to program EPROMs, and a null modem cable to connect to another computer's serial port, I can send you a specially modified ROM to diagnose the power on issue, which will definitively show what's actually running and where it's getting hung up. Contact me by Email (the PM notification Emails from on this site don't appear to work these days).
http://amiga.serveftp.net