I started with the fuel because I've seen what the modern gas does with old rubber parts and hoses. Other then replacing the filter, you have made no reference to doing anything else to the original fuel system. Fixing anything else and ignoring the gas and the old rubber issue is a disaster waiting to happen.

A 51 year old car with 51k miles on it has spent a lifetime sitting. Fuel systems deteriorate over time. Everything in the fuel system (in addition to the rubber) could be junk, including the tank, the fuel pickup in the tank, the steel lines, the fuel pump and the carb. A pinhole in a hose or steel line will give you what you are experiencing, and may not leak any gas at all, you only need a little air separating the gas to cause a stumble, and it can come from anywhere in the fuel system.

Old point style ignition systems are generally not effected by time, they usually work, or they fail completely, or there is an issue with wires or wire insulation breaking down. Who knows what rodents may have done to the wiring on that car in 51 years, they do like to chew on wires.

You have a 51 year old car with 51,000 miles on it, most were probably put on 20 years ago. Don't expect to jump in and drive it like you would a modern car with 51,000 miles on it. Unless everything has been kept up to date, things like brakes and wheel bearings, tires wipers, and a bunch of other stuff needs to be serviced simply because that stuff may be very old, or the next thing we will hear is "My brakes failed, what happened?" Gene