As it works on volts, then the fetron can't be the problem either.
There's not much it can be?, all it does on ohms is feed 1.5V DC through different value resistors to the input socket, and the differential amplifier (double triode) amplifies that voltage and feeds the meter.
So it's essentially just a potential divider, with the bottom half the resistance you're measuring.
As you can set the FSD using the pot, then the meter part is working fine, but presumably the resistor(s) from battery to input socket are high resistance somewhere, which means you're permanently on a high resistance range.
Have you tried reading a high value resistor? (in the mohms), and does that then not go to zero?.
Simple way to check, put an ammeter across the input socket, and set the VTVM to ohms and x1 range - the external meter should read 150mA, then try the x10 range, it should read 15mA, and so on as you go up the ranges. I suspect any current ranges will be extremely low?, which means it's working on a very high range.
Assuming it is very low current, try the external meter directly on the switch wafers - leave the VTVM set to ohms x1, and measure the current to chassis with the external meter. Pin 3 of range switch wafer C is the ohms output current, so should read 150mA - if that's OK, then try on the Function switch, pins 2 and 3 of wafer A. Pin 2 goes directly to wafer C, and should read identically to the previous test, and pin 3 goes to the input socket, and should read exactly the same as well as the wafer joins them together.
If the test for 150mA on pin 3 of wafer C fails, then try on pin 11, as that's the 150mA input from R30.
So measure those currents, and get back to us.
Really, it can only be the 1.5V battery, R30 (9.1 ohms) or either of the two switch wafers - the tests will prove which.