When you attempt to start a vehicle that has been sitting for years you can run into multiple problems. I’ll list the things that can be checked for free. Obviously, the fuel has gone bad by now and will need to be changed but the varnish it has left behind can be so bad that it plugs the fuel line and tanks outlet. It can turn to a solid and completely block the flow or restrict it. It can be so bad a coat hanger won’t go into the steel line. The entire fuel line will need to be replaced. I bought ’33 BB truck. I could not get anything to go into that steel fuel line, it was plugged that bad. I replaced the steel fuel line, and I had the tank boiled out at a radiator shop. I even sloshed bolts and nuts around in it and it still would plug the new line within one-block of driving. Here some good advice, always road test uphill from your house so you can coast back home again. Somebody is probably asking what finally fixed it? I sold it before I had a chance to find another tank.
My dads’ flathead ’32 roadster sat for fifteen years, and the fuel turned to muck. It would idle on new fuel but not drive to the end of my driveway without plugging the fuel line coming out of the tank. I have the tank out now. Beside the rust and varnish in there, there were a rusty beer bottle caps. It ran fine with them in there until the fuel turned to vanish. I’m pretty sure it also now has a stuck valve from sitting all those years, it runs like it does. It’s on my to do list behind many other Ford projects. The neighbors’ 1913 Model T, that last ran in the 1940s received a new engine. I was the only one that new how to drive it, so I got to do the maiden voyage. It ran for a few miles and then acted like it was running out of gas because it was. I let it sit awhile and enough fuel would gravity flow into to the carb and I could go a couple of blocks and then quit again. I had to hand crank it about thirty-times and by the time I returned my already bad back wasn’t working at all. That turned into an opioid day. That day, I got the urge to own an early Model T or any vehicle without a starter out of my system. I think I’d rather have the horse the Model T replaced. Same with his ’41 pickup that sat for a few years. It would idle but would not make it up the street. Same with the ladies ’34 pickup down the street that sat for years. I drove it to her new house, a couple miles away using a can of gas and an electric fuel pump. And yes, for some reason there is a strange number of flatheads living on this street with defective gas tanks. All true stories
You could be getting just enough fuel through the line to idle but not enough to drive it. You can disconnect the inlet line to the carb and crank the engine and you should see a good flow of fuel going into a glass container. Disable the ignition but not in a way a plug wire is going to create a spark. And always do this kind of stuff outside so you don’t burndown the garage. You can loosen the gas cap and take a rubber tipped blow gun and force air down the line and into the tank. This might dislodge anything in the line or tank outlet. This would only be a temporary fix. The tank would need to be removed and cleaned. A tank that is rusty inside is the one part I would consider replacing with a reproduction. It's almost impossible to get a nasty tank like new again inside unless you can have it chemically dipped.
What I do when I suspect a plugged line is I place a can of gas with its own low pressure electric fuel pump and filter in the cab and run a rubber fuel hose to the carb and go for a ride. It the problem goes away I know there is blockage in the line or tank, or the fuel pump is bad. It eliminates all that stuff as being the problem and only leaves the carb as the culprit. At least as far as the fuel system is concerned. Just make sure to cap off the outlet fuel line from the original pump so it’s not squirting fuel or your going to have a bad day. You can also leave the gas cap loose if you suspect it's not venting the tank. If the tank is not vented the pump cannot suck fuel out of it.
Once you have established enough fuel is available to the carb you need to find out if it is actually making it into the carb. Another recent example, a freshly built flathead in a friends AV8. It ran around the block for a while and then it ran out of gas. It was the neoprene needle and seats in these new carb kits sticking closed. Either they are not compatible with the new fuel, or they are just garbage to begin with, probably both. Solid metal needle and seats in both carbs fixed the problem. This is a very common problem.
Back when I was working on a fleet of 400 vehicles, they were all carbureted engines and point type ignitors. We did a tremendous amount of carb rebuilds and they always worked when we were done. The one thing a carb should not do is be allowed to dry out. That’s not a problem on a daily driver but it is on a collector car. We had a mechanic at work that had come from a Ford Dealerships. He had been the Ford dealer’s only tune-up man in the 1960s and 1970s. He was sharp on that stuff. He taught me a lot when I was starting out. He had a blow gun with a replaceable tip so he could screw a piece of 3/16 brake line into it. He used the 150 lb. shop air to blow out the passages. Then he took a can of carb spray with the straw nozzle and forced it against every orifice to make sure it sprayed out the other ends of the passages. When he was done, he knew the internals were perfectly clean. He checked the machined surfaces for flatness. The throttle shaft for wear and the linkage for binding. He installed a quality Hygrade brand power valves and accelerator pumps, new gaskets and it worked liked new. every time. The first thing he did after a rebuild was to blow into the carb with his mouth (do not use compressed air or you will crush a brass float like a submarine at crush depth) to make sure the needle and seat was open. While he was blowing, he’d flip the carb upside down and the needle would prevent him from blowing into the carb. One of the things he did that some people might not agree with was to spray all the gaskets with silicone. That prevented them from ever sticking to the parts. On Holley 4V carbs you can spend more time scraping gaskets off the metering blocks and bowls than you do rebuilding the rest of the carb. The silicone spray made the gaskets just peel right off. Because we rebuilt those Holleys almost yearly that was a neat trick. The phenolic base gaskets he coated with white grease so that the base gasket was reusable each time.
If you made it this far you shoud have fuel getting into your carb but maybe something is still not right. You mentioned you adjusted the air/fuel ratio screws all the way in (gently) and it did not kill the engine and it should have. That tells you the engine is getting fuel from somewhere that it shouldn’t at idle. Usually that is a leaking power valve, and it would be running rich which it sounds like it is. I have also seen carbs with cracks leak fuel into the intake, and of course float levels too high or excessive fuel pressure overpowering the needle and seat. With the excessive pressure fuel will be pouring out of float bowl vent and into the venturi. It will usually drown the engine and it will die. The only fuel the engine should be getting at idle is from the idle passages which are below the throttle plate, you cannot see them from the top looking in the carb with the plate closed. The fuel the engine is running on at idle is coming from below that plate. There should be no fuel at idle coming from anywhere above the throttle plate. If you have opened the throttle plate up to compensate for another problem. Enough air could start flowing past transfer ports and suck fuel out of them. Then adjusting your air mixture screws might not make a difference. The throttle plate should just barely be open at idle. Study the basic circuits of a carburetor so you understand under what conditions each of them flow fuel.
Here is where your carb gets a little confusing to me. You also said you had to pull the choke half closed to get it to idle. What is happening when you do that is you are creating an artificially rich condition to get it to idle. The first thing you did with the mixture screws suggested the engine was too rich and then the next thing you did with the choke suggests it’s too lean. I want you to take a propane bottle, carb spray or starting fluid and just add a little into the carb as it is idling and see if it straightens out or if it makes it worse. This will tell you for sure if the engine is not getting enough fuel or it’s getting too much at idle. You can also crack open the propane bottle just a little and run it around the intake, carburetors’ base and throttle shaft.. If the idle changes you have found a vacuum leak. A propane bottle or map gas is fantastic tool for checking air fuel mixture problems.
You mentioned it backfires. If it is backfiring through the carb that can be a sign of a lean condition. A non-operating accelerator pump or one positioned to squirt a low amount of fuel would cause a lean condition resulting in a hesitation and possibly a backfire through the carb, not the exhaust. As the throttle is moved you can easily see it squirt looking down the carb. You do not need to have the engine running. The only time the accelerator pump would cause a lean condition is while the throttle is opening. Once it has opened and is no longer moving the accelerator pump is no longer adding fuel and would not be the source of any problem under steady driving
You said, you had a fluctuating vacuum gauge, that can indicate a valve problem. Diagnosing it depends on the rpm and speed at which the needle fluctuates. Each time the piston goes down on the intake stroke it creates a vacuum (suction). When all cylinders are creating the same vacuum, the gauge is steady. If one cylinder is not creating the same vacuum the gauge momentarily drops when it’s that cylinders turn to suck. If the engine sat it could have a valve sticking. I got one in my garage right now that I have not properly diagnosed with a vacuum gauge, Just the sound of the engine was enough. If it’s an intake valve that happens to stick open the piston will push the mixture back up into the intake and disrupt the incoming flow to the other cylinders. It will run horribly, which the one in my garage does and mine has a loud tapping lifter, another sign of a stuck valve. It ran perfect when it was parked, and the lifters were quiet. A sticky intake valve is my number one suspect. You did a compression check which showed the valves were closing so you do not have stuck open valve, at least not at cranking rpm. It could be intermittently sticking while running. Multipole problem can really mess with your head, but multiple things can and do go wrong on vehicles.