Well you said it was "flooding out". You have to determine if that is actually happening OR if something else is causing it to die. Does gas run out of it when it "floods out"? Does it have normal compression when it dies? Is the gas cap vent open? Is there spark when the engine dies? These are all things you need to find out.

Walt Conner

With the air cleaner removed, I see gas flooding into the air cleaner housing, black smoke pouring from the exhaust. The engine begins to "bog down" and eventually is running so rich it cannot keep going. Spark appears to still be good because the engine keeps firing until the end; I haven't checked the compression yet-I suppose that would be an indicator of a valve problem. There appears to be gas in the oil as well.

The "valve issue" here is in the carb needle valve.
Gas in the oil is likely due to the carb needle valve failing to shut off the fuel flow when the engine is not running. I'm guessing this machine has a gravity feed fuel system (fuel tank higher than carb).
The gas in the oil can be adding to the "flooding effect" as the gas in the oil will vent out the crankcase breather and into the engine intake manifold.
Install an inline fuel shutoff valve and keep it turned off at all times when the engine is not running.
Change the engine oil at once.
The carb needle valve can be tested effectively by removing the carb from the engine. Take the float bowl off the carb.
Attach a piece of fuel hose about 18 inches long to the carb fuel inlet fitting. Mix about a cup of water with some dish washing detergent or "bubble bath".
Turn the carb upside down in your hand and pour a generous amount of the bubble solution into the upside down fuel bowl cavity right where the needle valve is located.
Now, put the open end of fuel hose in your mouth and blow hard into the hose. Watch the carb needle valve area for soap bubbles to appear. If you see any, press gently on the float to seat the needle better. If the bubbles contiue even with a slight finger tip pressure on the float, your reconditioned card did not get much conditioning before you bought it.

I understand your theory with the needle valve, but this problem has been happening with BOTH carbs (the original and the rebuilt). Knowing that it is a common problem, is it really that likely to be the exact same issue with 2 different carbs in a row?

Well, ponder for a moment WHY that "reconditioned carb" was available for sale. It must have been a problem for somebody else (or they thought it was a problem) or it would not have been to rehab.
Now I can't answer exactly what the odds are of 2 carbs in a row having similar defects, but I do know that the symptoms you describe fit exactly the symptoms of a needle valve and/or float problem (leaking or flowing fuel when it should be sealed tight).
Is this fuel system gravity fed or does it feature a fuel pump?

This is a gravity feed system.

The carb was purchased from a business that salvages and reconditions mower parts-supposedly with OEM parts and tested as far as possible, so while anything could happen, this isn't a case where someone simply yanked an old carb off an engine and sold it to me.

Obviously the float/needle is not cutting off fuel flow, but the question is why? If the float and needle themselves are good, what else can make them not work right? Can excessive back pressure from the engine prevent the float from rising enough to close the needle?

No, back pressure has nothing to do with the float. The float works exactly like the classic "ball float" in the toilet tank that rises up with the water level and cuts off the water at the precise level in the tank that it was set for.
The float in the carb works the same way.
If the carb has a brass float, these can become perforated and fill with gasoline. If that happens, it is no longer a float, it is a sinker.
If, somehow, that carb was dropped before you bought it (even if it was already in a package) the float arm could have been knocked out of adjustment, and will not now exert enough lift pressure on the needle to seal it tightly to the seat surface.
The hinge pin that the float rides on could be binding and not allowing the float to exert enough lift. With the bowl removed from the carb, you should be able to lift the float with a finger tip and then it should fall back easily and completely when the finger is lowered.
The soap bubble test would determine if the needle valve was shutting off or not.

Float the float in a coffee can of two inches of gasoline, and see if it sinks! Might take some time for this to happen, like several hours, but if the float sinks, there is your problem. It won't do its correct job of shutting off the flow of gasoline, which can meander over into the crankcase of the engine, resulting in a gassy smell of the engine oil. Do be careful with an open coffee can of gasoline!

OP - one other important thing: pull the engine's oil dipstick and smell it for presence of gasoline. If you smell it, DRAIN THE CRANKCASE OIL, refill with fresh & replace the oil filter BEFORE running that engine again.
Raw fuel in the oil will not provide proper engine lube, possibly leading to total engine failure.