There is an E-series maintenance manual here:
http://matthieu.benoit.free.fr/elan/E series Maintenance manual.pdf.
I am not sure how close that is to the E9A.
It states there is a Z80 and gives a lot of technical details. I am still looking for the schematic however...
Nothing 'serious' is present in the battery-backed RAM that would cause the device not to function when turned on:
I wonder how close this schematic is to your machine:
http://matthieu.benoit.free.fr/elan/EA series schematics No 285.pdf?
It may suffice to debug the CPU section.
Perhaps post some high-resolution photographs of the main logic board so that we can compare your board with the schematic.
It looks like an ICM7218D is used to drive the seven segment displays. This is an 'autonomous' device and doesn't have a reset pin. As a result, what you could be seeing on the display is the 'random register values' when the programmer powers up. This will also explain why you get largely consistent results, but (occasionally) the odd difference. My starting guess is that the Z80 (or whatever) microprocessor is not starting up correctly and initialising the display driver IC.
I would check all of the power rails to make sure they are correct, the clock pin and the reset signal. The reset signal appears to be just a CR circuit. If this is the case, the capacitor may have degraded and there is no adequate reset signal to the microprocessor. After that (assuming it is a Z80 microprocessor) I would check the /M1, /MREQ, /IORQ, /RD, /WR and /RFSH pins for any signs of activity.
Dave