I thought about this subject a little more and I have a couple more suggestions. You have two separate symptoms, a carb that looses fuel and a pump that at cranking speed won't fill the carb back up. I‘m pretty sure you have two separate problems.
First, as far as the carb loosing fuel, there are only two places the fuel can go. If it sits too long it can evaporate out of the bowl. This can leave a varnish in there that can cause the needle and seat to glue its self shut and not allow any fuel to flow into the bowl when you go to crank it. I used to have this problem on some of the GTE/Verizon trucks that were not being used. They would sit for months between services and the needle and seat would get stuck in the closed position. Someone would have to tap on the carburetor next to the needle and seat while someone else cranked the vehicle and then they would usually fire right up. Since your vehicle is not sitting for months this is probably not your problem.
Second, the fuel leaked out of the carburetor and ran down inside the intake. This could be caused by a faulty internal part, gasket or even a cracked or warped carburetor. I have seen carburetors crack and warp but I have never heard of it being a problem on an early Ford Carbs. If I suspected the carb was leaking I would remove it and fill the bowl with fuel to the proper level and then see if any fuel leaks out the bottom of it, then you can see exactly where it is coming from. These early Ford carbs are nothing more than glorified lawn mower carburetors. If you can fix your toilet bowl float your probably qualified to rebuild one of these carburetors. Just don't over tighten anything, snug the little screws down evenly and when they stop turning STOP TIGHTENING THEM! I hate working on carbs that someone else has worked on, usually half the screws are stripped out.
As far as the fuel pump not priming the system. There are check valves in the fuel pump. The inlet valve will open on the vacuum stroke to let the fuel flow into the pump then it closes on the pressure stroke so that the fuel is pumped to the carb and not back into the tank If one of these check valves were to go bad and stay just slightly opened, at slow cranking speed the fuel might just get pumped back into the tank. With the engine running the pump would be moving more fuel and not all of it would be able to flow through the slightly stuck open check valve so the engine gets enough fuel to run
A couple of other issues The old cars had rubber hoses with woven cloth coverings. If your restoring your car this is the look your going to want. I had a ‘57 T-bird that my mother had bought brand new, she had it for two years and then traded to my grandfather for a small house. In the seventies he started worrying he would not be able to get parts for it so he went down to Ford and ordered every rubber part you could possibly think of for the car and stored them in air tight peanut butter jars. Around 1988 he died and I inherited this low mileage original paint ‘57 T-bird that had not been started in about ten-years. I rebuilt the original holley carb, installed a new fuel pump but I could not get the pump to prime the carburetor. If I filled the carb up with fuel it would start and drive all day long. If I let it sit for a couple of weeks I could not get it started again with out first filling up the carb bowls with fuel.
Here is what I eventually found. You always need to check the rubber fuel lines between the tank and pump for cracks. At one time or another we have all gotten straws with cracks in them, what happens when you suck on the straw, nothing, no fluid comes up the straw, all you get is air. Your fuel pump works exactly the same way. My ’57 T-bird had a short piece of hose at the back of the tank and one of those woven cloth hoses at the fuel pump. It was a strange little hose with factory brass fittings crimped on each end. I removed the hose and soaked it in dish washing soap and blew through it. Bubbles came out of the cloth the entire length of the hose, there was my problem, the fuel pump was sucking air. I wanted to keep this car absolutely original and fortunately for me my grandfather had bought a new fuel hose and store it in a peanut butter jar (he must have eaten 55 gallons of peanut butter a month) I pulled out this NOS hose with its nice original Ford part number tag still wired to it. It looks just like it was made yesterday, nice and flexible, the brass fittings had not even tarnished. I stuck it on the car and I have the same exact problem. I pulled that NOS cloth fuel line off and poured dish soap over it and blew into it, same thing bubbles from one end of it to the other. I can’t tell you how many time I have installed bad new parts, it screws with your head. I would have thought that there would have been fuel dripping from both of those hoses but if never showed any sign of leaking. The problem with those cloth hoses is you cannot see what the rubber looks like underneath. If you have to use them for a 100 point correct restoration, I would replace them every few years.
Another thing to consider on flatheads is the fuel pump mounting stand. The bases warp on them and this reduces the stroke of the fuel pump rod. I have no idea what the stroke of a flathead fuel pump rod is but lets just say it‘s a 1/4 inch. I have bases with an 1/8 inch warp in them, in fact 90% of them have 1/8 inch warps in them. it’s really hard to find a stand with a nice flat base. The fuel pump cam lobe could also wear and reduce the stroke of the fuel pump rod. I would check the base of your fuel pump stand to see if it is warped and while you have it off measure the travel of the fuel pump rod.
I bolted to fuel pump stands together and took a picture so you can see how bad they warp at the base.
Jerry