IIRC, and it has been awhile since I have been inside one of these, there is a 3 terminal 5 volt regulator that supplies power to most is not all of the LEDs and the BCD decoder integrated circuit for the station indicator LED'S and possibly other things.
This same 5 volt regulator also supplies, via a separate diode isolated line, voltage to the same place as the backup batteries.
There is another three terminal voltage regulator, 12 volts.
Both of these regulators are supplied by their own fuse.
Simple question, but I have to ask, because not everyone does this, but did you remove the fuses when you checked them with your DMM?
If the fuse for the three terminal voltage regulators is bad, no LEDs. If it is good then on to other things.
Again, IIRC this unit has a number of power supply voltages.
The protection relay, if that is what you are hearing click, is driven from the positive 35 volt power supply rail and will turn on even if other voltages are missing, because it is biased on from the same 35 volt line that drives the relay. It is turned off if there is DC voltage on the speaker output or if there is excessive voltage drop on an output transistor emitter resistor (excessive power output).
It is not a bad thing to start with measuring all of the power supply voltages.
BTW, the backup battery supply is diode isolated from the 5 volt regulator and diode polarity protected for the battery back up voltage. Unless the protection diodes are bad or leaky, batteries, no batteries, weak batteries or batteries installed incorrectly should not cause a problem.
If all power supply voltages are present and correct, then it is on to other things.
An addition, again from memory, but if you select tape one and the 5 volt regulator is doing is job, the tape LED should light. IIRC this is just a connection to ground on one of the terminals on the tape selection switch.
Power up the unit and push the tape 1 button and see if the tape monitor LED comes on. If it does, then at least one of the 5 volt lines from this regulator is working.
If loopstick can post the complete service manual with voltages and parts values that will help.