It sounds like you've done a thorough job with the cleaning and lines, but the short run-time is definitely frustrating. To answer your main question: Yes, if the bowls were properly primed, the engine should run for significantly longer than 3–6 seconds.
Even at a high idle, a Carter AFB with full bowls should typically keep an engine running for 30 to 60 seconds, if not longer. Since it's dying almost immediately, here is what is likely happening:
1. The "Vent Tube" Priming Issue
Pouring fuel down the large brass vent tubes is the correct way to reach the bowls, but it can be deceptive.
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Air Pockets: If you pour too quickly, air can trapped in the bowl, making it seem full when you've only actually added a tiny amount of fuel.
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The "Throat" Prime: When you pour fuel down the throat, you are essentially just using the carb as a funnel to the intake manifold. That fuel burns off in about 3 to 5 seconds—which matches your current symptoms perfectly. This suggests the engine is running only on the fuel you poured down the throat and isn't drawing anything from the bowls at all.
2. Accelerator Pump Test (The Best Diagnostic)
Before you swap that fuel pump, do this quick check to see if there is actually fuel in the carb:
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With the engine off and the air cleaner removed, look down the primary (front) barrels.
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Manually move the throttle linkage quickly.
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What to look for: You should see two strong, solid streams of fuel shooting from the accelerator pump jets.
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The Verdict: If you get no squirt, the bowls are empty. If you get a weak squirt or just air, the fuel level is too low to reach the pickup.
3. Why the Fuel Might Not Be Reaching the Barrels
If you are certain you put enough fuel in those vent tubes (usually about 2–3 ounces per side) and it still won't run, check these three areas:
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Stuck Needles and Seats: Since you just rebuilt it, it’s possible the needles are stuck in the "up" (closed) position. If the needles are stuck shut, no fuel will enter the bowls even if the pump is working perfectly. Gently tap the top of the air horn near the fuel inlet with a screwdriver handle to see if they drop.
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Float Adjustment: If the floats are set too low, or if they are catching on the side of the bowl/gasket, they won't allow the bowls to fill or the jets to pick up fuel.
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Vapor Lock/Fuel Pump: You mentioned replacing the pump. If the pump is weak, it won't be able to push past the initial air in the lines. Since you cleaned the tank and lines, ensure there isn't an air leak at one of your new rubber hose connections on the suction side (between the tank and the pump). A tiny pinhole leak there will cause the pump to suck air instead of pulling fuel from the tank.
Next Steps
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Try priming the vent tubes again, but do it slowly.
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Perform the Accelerator Pump Test mentioned above.
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If you get a good squirt but it still dies, you may have a massive vacuum leak (check the large power brake booster hose or the carb-to-manifold gasket).
If you find the bowls are bone dry after that test, then moving forward with the fuel pump and checking for suction-side air leaks is the right move!
