Definitely not "a Mopar thing". All gasoline engines need spark and a fairly narrow range of O/F to fire. Mopar never made carburetors or fuel pumps. GM & Ford used same suppliers.
Could be a lean idle condition. Most likely from a major vacuum leak. A broken PCV valve could let in a rush of air, as RedFish says. Another is the vacuum hose to the brake booster (plug it). Also the carburetor gasket, in which case you might hear a sucking sound. I agree with you that it should run fine off the battery at any rpm. The alternator's job is to keep the battery charged.
To get an idea. On a newer fuel injected engine, remove the big hose to the brake booster. The engine will greatly speed up. Most will run fine since the fuel controller measures the increased air flow and compensates (perhaps not if it uses a mass flow sensor and the booster connects downstream). However, the carburetor in your classic car will not see this increased air flow from the leak, so will not add the extra fuel needed, thus the engine will misfire and shake, if it runs at all. A classic test is to flow propane around any suspected leak. It the engine smooths out, you found the leak. Don't smoke.