What could cause the old gas furnace to keep cutting out

It's common for HVAC systems to be improperly designed. It's also a common misconception that it's okay to close vents. If a regular density filter is restricting airflow (and causing the unit to heat up too much and trip a safety switch) then you need to do something about the airflow.

In general it's not a good idea to close vents because it changes the "static pressure". I don't know the details but basically you don't want the pressure in your vents to be much higher than the pressure in your house. If so the fan is working too hard and there isn't enough air passing over the heat exchanger.

Check the high limit switch. If the furnace is allowed to continuously over heat, It's likely the high limit switch has burned out.

Make sure that the outlet vents are clean, and clear of debris. You need good air flow both in, and out of the system. also make sure the heat exchanger is clean, so it can transfer heat properly to the passing air.

If the thermostat has an option for Fan, switch it to Fan and make sure the fan starts. If the fan does not turn on, you should contact a trained HVAC tech. Capacitors can be deadly if not handled properly. You could accidentally discharge the capacitor into yourself, leading to electrocution. Or short the capacitor, leading to an exploding capacitor. In either case, it could be deadly.