If you can do some work on your own car, you might want to take the fuel line off from the sending unit under the tank, and also take it off at the inlet to the fuel pump and add some air pressure to test your theory. If you don't have an air tank, a good blast from your lungs will be sufficient.
Just remember to blow out and not suck in, or else get a mouthful of gas. Not very tasty.
If that is clear, then take the line off from the outlet of the pump and the carb and blow toward the carb. Now, if that is clear, you have to know that you have a filter in the gas tank of these old dinosaurs. It's called a sock. You must first drain all the gas out of the tank and take the sending unit out. My 65 has a plug on the bottom corner of the gas tank. Hopefully yours does too. The sock is attached to the sending unit. The tank may have rust particles in it, and the sock is clogged, not allowing enough gas to get to the pump. If that is the case, I would replace the entire sending unit with the new sock. And, if that is the case, you may want to consider replacing the gas tank in the near future. You are right about some mechanics won't touch a carbureted car, but some will. I just had a local mechanic who I know well to work on my 1981 F-150 with a 351 Windsor with a Motorcraft 2100 carburetor. My problem was vacuum line problems though.
You are just going to have to realize that in order to drive these dinosaurs, you are going to have to do most of the work yourself. I have been working on my 65 for almost 20 years now, and still learning.