Could be several things. You will have to do some testing. Assume that you have a fuel pressure gauge located at the carb to know that your pressure is 3 & 2 pounds?
Your fuel line should be 3/8" and should be OK. You don't need 1/2" fuel line unless you are pushing big HP numbers or spinning high RPM's. If you were to go with 1/2" fuel line, then you would most likely need the electric pump at the rear to push the gas to your mechanical fuel pump.
First thing, often times the wrong gas cap has been installed on the car. As I recall, the '67 needs the vented cap IF it does not have the vented tank. The wrong cap can create a vacuum and the fuel pump will starve for gas. So first test is to remove your gas cap and run the engine up to see if this corrects the fuel pressure. If no change, next test.
Next I would disconnect the line that goes from your tank to the fuel pump. Then install a fitting that will allow you to run a long length of rubber hose to a gas can. Fire up the engine and see of the fuel pressure on your gauge comes up. If it does, then the problem is in the fuel system from the tank to the fuel pump. What come to mind next is the "fuel sock" found at the end of the tank pick-up tube has become clogged or collapsed - so the fuel pump is straining to draw gas from the tank.
Check ALL your fuel lines. ALL of them should be replaced with contemporary fuel lines compatible with ethanol laced fuel. Ethanol fuels destroy the older rubber lines. You could have a collapsed rubber line or even a cracked line sucking in air. Check all your lines and replace as needed.
If your fuel pressure does not go up any by doing the gas can test, then the issue is from the fuel pump to the carb. Your fuel pump may simply be bad and needs replacing. Is your fuel filter going in the right position? Some fuel filters are directional specific and if you install them backwards, you won't get the needed flow and some won't flow at all (Silly question, but I gotta ask).
Vapor lock can be a problem with ethanol laced gas when the engine gets hot. You may want to add a gas return line which can help with this problem IF your car does not have one already - it can be added if you don't.