More problems...
1) The output level potentiometer is very "scratchy" and cuts volume at certain points.
EDIT: required disassembly and cleaning: FIXED!
2) It appears I re-soldered wires to the VU level meters on backwards during the earlier main PCB corrosion repairs.

The meters are moving in opposite direction! LOL!!
EDIT: FIXED! Now the VUs work.
3) The "take up function at loading" is not working.
But maybe this VERY early 580 has no "take up function at loading." It also has no pressure pad lifter and no fancy right pressure roller assy.
EDIT: C428 does not exist on the logic board for this early 580. So no "take up function at loading." Guess I never noticed that until now! I'm more used to the 480.
4) Dolby button is very "scratchy" when pressed and then the music peaks have "scratchy" noise. Dolby out is ok.
EDIT: Dolby switch required contact cleaner. Now no noise from the switch. But the RIGHT Dolby IC likely has damaged op amps (very bad distortion on music peaks!) and needs to be replaced. The left channel is fine, but the Dolby ICs are matched and must be replaced in pairs.
5) Lots of static in REW! (I know the reason... )
EDIT: This was because the MUTE signal wire from the Logic PCB was disconnected to get audio. See posts below. CORRECTED by reconnecting MUTE wire from Logic PCB.
6) Two VU bulbs burned out...
7) STOP and PLAY button bulbs burned out.
EDIT: now those bulbs have been swapped for the FF & REW bulbs.
Replacing the STOP bulb fixed the original no audio issue for this thread!
And:
Does LINE OUT work? (Untested)
EDIT: YES!
Does REC function work? (Untested)
EDIT: Yes!

So the 580 is currently working better than the 480!
Record levels are very slightly unbalanced (1-2dB).
Now to see if REC will still work after removing the diagnostic jumper from R311? (Or maybe just leave it and remove completely the other jumper?

)
Thanks again to
dankik for all the help getting this far
Note to
dankik and for future reference;
I now also know this is a VERY early very low S/N 580! Probably one of the last made of the few from the initial revision. The main PCB is the earliest "A" board. The boards shown in the manual are "B" ("previous 580" in manual) and the later one. That's why it's quite different.