I see the frustration here, and personally, I get more frustrated with things that don't make sense to me.
You stated that you know little about carburetors, but that the old Toyota starts fine. There have been some good responses so far, but to recap:
The Toyota uses a carburetor that is sealed to the environment and the chance of fuel evaporating in diminished. It is obviously less sensitive to the modern fuels that we all buy at the pump. The 750 Holley is a great carburetor, but yours seems to run dry after a few weeks. I have dealt with this with every old car that I have had. It so happens that every old car has had either the factory late 60s/early 70s air cleaner and emissions package or an simply an open element air cleaner. To me, this one factor seems to prove the theory of fuel evaporation. The Demon 850 in my Charger poses no startup problems until about 3 1/2 weeks or so. The 67 Dart uses a 750 Holley on a 360 and acts about the same. I have ThermoQuad carbs on 3 other vehicles. They range from dry in a week to easy starting after a month of sitting.
The solution? As others have stated, the electric fuel pump is one way. Pouring fuel down the throat is cheaper though.