How to debug a precision temperature controller?
What you have there is fairly ancient, I would guess 1970s, so around 40 years old, give or take. Maybe 50. 

The temperature probe can easily be replaced by a resistor for testing. You will need to figure out the connections. Assuming it's a platinum RTD it probably has a Kelvin connection meaning that two wires run to each side of the element. The resistance is probably 100\$\Omega\$ at 0°C but the coefficient could be 38.5 or 39.2 (older US-made instrumentation was sometimes the latter). Chances are the sensor is okay unless it is physically damaged. 







simulate this circuit &ndash; Schematic created using CircuitLab

Something that old, and from your 'buzzing' comment, my first thought would be that the electrolytic capacitors (particularly the power supply capacitors) have dried out. There are only 5 of them (the green cylinders) and they can be replaced with new ones for a few dollars. Polarity is important, as is capacitance and voltage rating. Take detailed photos and get someone else familiar with electronics to do it (check the parts, do the de-soldering and the soldering) preferably, because if this is screwed up I guarantee nobody is going to want to put much effort into fixing someone else's mess. Just the way it is, sorry. 

The other parts are probably about as good as new. It looks like wirewound custom resistors on the PCB- maybe not but hard to see. I think I see a dual-secondary gate-drive pulse transformer (the red object with 6 leads) which probably drives two SCRs on that homemade heatsink. 

Functionally this controller probably controls a single heater with the 15A output- and probably via phase control. It's not clear why it would have both a gain and a proportional bandwidth adjustment as normally one is considered reciprocal to the other.