<snip> it ran perfectly fine just after I bought it. However, after running it for a summer, it starting losing power and it was having a hard time starting and a hard time staying running and it would want to die out when it got hot, etc... So, that was when I parked it figuring it needed a good carburetor cleaning. That was about 2 years ago now. However, the engine DID run fine when I first bought it back in 2013, so that would lead me to believe that the engine internals should all be good. But I do know what you mean about it backfiring through the carburetor and I'm not sure what is causing that. These are classic symptoms of compression issues or ignition failure.
I am getting good spark. I had to replace the spark plug wire though when I removed the gas tank as the stock wire was routed through the gas tank mounting bracket somehow and I couldn't get it out in one piece, so I had to cut it and use a new plug wire that was longer. Did you replace it with wire-core plug wire, or carbon-core resistance wire?
Since I can't easily remove the points cover without removing the whole engine, I have not physically checked the condition of the points. However, like I said I get good spark and I checked the timing on it with a timing light as per the instructions in the service manual and it seems fine. OK, there. However, if the points were bad or out of adjustment or the coil was bad or the spark plug wasn't firing right, etc. then I would think it would not run at all. Not a correct assumption. .....The point adjustment will affect ignition timing, so if the timing is correct, the adjustment is OK. .....However, a failing condenser, or oil on the points, or burnt points can still cause problems. .....A failing coil, a bad connection, or a faulty ignition switch can all cause intermittent spark.
If you can put an inline visual spark tester between the plug wire and the plug terminal, you can verify the spark is constant while the engine is running, cranking, or dying out.
As you can see, when the engine starts getting enough fuel mixture from the carburetor, it looks like it runs great, right up until it dies. It definitely seems like a fuel issue to me, not ignition/spark. Have you tried to add more fuel by opening the main adjustment ?
Too high a float level can cause flooding. ...Too low can cause starvation. ....If it is close, this shouldn't be a problem since gas will enter the carburetor bowl faster than the engine can burn it.
Maybe the valves are sticking, especially from sitting. Not a common occurrence on those Wisconsin engines. I have no idea how to really test that though unless I start taking the engine apart. But again, when it does start running okay, it seems to run so nice and smooth that it seems unlikely that a valve would be sticking or something. A burnt valve, or a valve not seating can create compression problems, low power, hard starting, or spitting through the carb (intake valve).
I'm not sure if the venturi was removed on this and installed upside down or not. Would that make it not run at all though or just run poorly like it is? I don't think I had the carburetor that far apart when I soaked it though, but it is possible that it could have been installed improperly by a prior "mechanic"... I believe from the parts list of the Zenith carburetors in the back of the Wisconsin TRA-10D service/parts manual it does show the venturi as a separate part. If you didn't have the carb completely apart when you soaked it, you may have done more harm than good. .....I haven't had one of those carbs apart in many years, so I don't remember if the venturi can be installed upside down. ...However, if a venturi is installed wrong, fuel is not siphoned as easily.
How do I turn off this damn auto-correct? --- It wants to keep changing venturi to venture?