Here's what I can tell you from looking at those pictures. I think you've probably figured it out already.
The device you've marked "thermistor or ceramic cap" is probably a PTC thermistor - I can see the tops of the letters "PTC" just to the right of it in the first photo. It's probably a "Polyswitch" or similar - a resettable fuse that protects the circuit against excessive current by getting hot and going open circuit. When the overload is removed, it cools down and reconnects the circuit. So it's called a "resettable fuse".
The four-pin connector JP2 seems to carry the positive supply voltages. (But not the 0V rail!) The BAT+ wire (positive from the battery, I guess) connects to pin 2. Pin 3 connects a supply voltage into the switch and it comes out on the middle pins of the switch, and goes out on pin 4. Pin 1 comes from the centre contact of the 4.5V DC connector position. When the switch is OFF, the resistor (brown-black-red I think - 1 kilohm) connects pin 4 of JP2 to the 0V rail, to kill the circuit.
Edit: To clarify, I think the pins on JP2 are:
1 = positive supply, before ON/OFF switch, from centre pin of 4.5V DC socket (not fitted), going to motherboard.
2 = positive supply, before ON/OFF switch, from red wire from battery, going to motherboard.
3 = positive supply, before ON/OFF switch, coming back from the motherboard, going to the ON/OFF switch.
4 = positive supply, after ON/OFF switch, going to motherboard.
The negative wire from the battery, and the outer contact from the 4.5V DC connector position, connect through the PTC (see above) to the 0V rail of the circuit board, which connects to three pins of the MIDI socket (probably unused), and possibly to the 0V connection on the micro USB connector, but apparently not to anything else. I would expect to see a connection from that rail to the motherboard. Do you know where this is? I don't think it's on pins 1, 2 or 3 of JP3; I think these come from the micro USB connector.
Pins 4 and 5 of JP3 are probably the MIDI data output from the motherboard; this is a current loop signal that feeds an optocoupler at the other end. I assume this unit only generates MIDI and does not receive it.
Pins 6 and 7 of JP3 connect to the two brown wires. What do these connect to?
So your red and green dots just show what connects to what, right? They don't show whether voltage is present t those points or not?
Yes, if that switch is an ON/OFF switch, I would expect positive battery voltage (fed to the motherboard on JP2 pin 2 or pin 1) to come back from the motherboard on pin 3. If you have voltage on pin 2 (from the battery red wire) (which presumably feeds into the motherboard) but no voltage on pin 3, I would say there's a break in the circuit at the motherboard.
It's possible that this break is deliberate; obviously, there is SOMETHING connected between those points at the motherboard end, otherwise there would be no need for those two wires. (Even if pin 2 was used as an "always ON" supply to back up some stored parameters, which is possible, there would be no need for pin 3, unless there is some kind of interrupting circuit between pin 2 and pin 3.)
Is there a switch on the motherboard to select between battery power and power from a power supply plugged into the adapter socket? If that's set wrong, you'll get continuity between JP2 pins 1 and 3, but not from 2 to 3, and the unit won't receive power.
Can you post photos of both sides of the motherboard in the area around where the wire from JP2 plugs into it?