A carbureted engine without sophisticated electronic mixture controls capable of adjusting the fuel mixture in a VERY narrow band of operating range (0.5 volts +/- 0.2 volt on an O2 sensor) several times a second CANNOT run a catalytic converter. The converter will experience a catastrophic meltdown within a very short time span.
Your C15 had an electronic controlled carburetor that used a computer to adjust the fuel mixture 10 times per second, probably a Rochester Monojet. It also had an O2 sensor that provided feedback to the computer once the sensor reached 600 degrees F. and started generating a signal.
Your description of the operation of a catalytic converter is correct for reducing emissions. However, speaking about GM specifically (seems appropriate for this site), the first vehicles with catalytic converters were built in 1975. The first vehicles with an electronic carb (Rochester) didn't show up until 1981. So, six years of traditional carb/catalytic converter vehicles. I don't know the specifics for Chrysler and Ford. All the car companies had to have converters in '75.
You're absolutely correct that converters will melt down if given enough fuel and oxygen. Everyone has a story about having a converter get too hot. Running consistently a little rich or lean isn't too big a deal from a heat standpoint. But, if you cycle the two from one extreme to another things do heat up. For example, something like consistent cylinder misfires that dump unburned fuel and O2 down the exhaust pipe.
You certainly can install a converter on an older vehicle. Regardless of the potential emissions benefits, it'll definitely burn off any excess fuel that might be causing a bad exhaust smell. However, you'll also want to make sure it's installed in an area reasonably close to the engine but away from anything that can't handle some heat (like fuel lines). It would probably be worth adding some sheet metal heat shields to the floorboard to protect it from getting too hot as well.
