Hesitation

That sounds typical of a problem with the accelerator pump. Assuming the pump is new, first sight down the center of the carburetor with the engine off, work the throttle, and watch for two nice streams of fuel. If they are weak or missing check for proper placement of the little metal check balls.

If the streams of fuel seem strong, pinch the rubber fuel line feeding the fuel pump, then stop the engine within about ten seconds. That will be enough time to remove the pressure in the fuel line to the carburetor and will prevent more fuel from going into the float bowl. Remove the top cover and check the level in the float bowl. If it is low compared to the gauge in the kit the fuel will have a hard time being drawn in when the throttle blade is open. The clue to low level is the engine will sputter and hesitate even when you open the throttle slowly as in when trying to go faster and faster on the highway. If the accelerator pump is the problem the engine will run fine at wide-open-throttle as long as you reach that point gradually. It's the sudden opening of the throttle where the fuel can't get moving fast enough when the accelerator pump is weak.

Just to be sure we aren't overlooking something stupid, do you have the air cleaner housing in place? What you're describing happened to me on a car I borrowed in the mid '70s. The owner's kid forgot to put the housing back on. At anything over about 3/4 throttle the hood silencer pad got sucked down and blocked air flow into the carburetor. Sometimes it suddenly started running fine on its own, and sometimes not until I opened the hood. I finally figured it out by slowing closing the hood and I saw the pad drop down.

The clue there was engine speed, not the rate of increase. If your problem only occurs when you snap the throttle open very quickly from mid-way, you may be running out of accelerator pump. Hold the throttle halfway open, THEN check the fuel streams when you run it wide open. If there's no fuel or a real weak spray from halfway on, the accelerator pump may be not taking a long enough stroke or the pump may be not fully filling with fuel. That would be a result of the level too low in the float bowl.

Is it possible there's a restriction in the fuel supply and the fuel is not refilling the float bowl fast enough? Can you hit wide-open-throttle on the highway if you approach it slowly?

That rules out an accelerator pump issue. The best I can suggest is if it spits and sputters when it cuts out it is most likely related to a fuel volume issue. If it cuts out suddenly and always at the same throttle blade position, it is more likely related to the nozzles and jets. A point is being reached where the air flow is not creating enough vacuum to draw in the fuel. I know that isn't helpful, and the only thing I can think of is to look at everything that is involved in drawing the fuel into the air stream. Were there two almost identical gaskets with slightly different holes, and you used the wrong one? Are the metering rods opening all the way?

The next thing I would do is repost your question so the rest of the guys will see it. Hopefully one of them will have a better answer.

Dandy! I gave some thought to a vacuum leak but they normally cause problems at low speeds, mainly a high idle and hesitation when accelerating.

If that line runs to the right rear of the transmission, that is connected to the "vacuum modulator valve". That is what sets the shift points. Under acceleration you will have lower vacuum. That tells the transmission to delay the up-shifts until you reach a higher speed. When you're slowing down with a closed throttle, the vacuum will be high and the down-shifts will occur at much lower speeds.

If you ever run into that hose sliding off repeatedly, suspect the modulator valve is leaking. The transmission fluid will rot the rubber hoses and make them mushy.

Chrysler only used a linkage to adjust the throttle valve in the transmission. That also adjusted line pressure so under harder acceleration the clutch packs had more pressure applying them to reduce slippage and make a firmer shift. Ford used some of each. I'm not a Ford expert but I know you have to look to see if they use a modulator valve or a linkage.

Very happy you solved this and was not a serious problem.