Squirt some pre mixed fuel directly into each cylinder, and rope it over… if it back fires, and you are gonna hate this, switch the plug wires and do it again. If that fixes the backfire then try to run it. I can’t think of any timing problems that might cause this, I can only say the only backfire issues I have ever had were from plugs crossed. Now that being said, what are the carb needle settings? You might be getting a lean sneeze and thinking it is a backfire..

Had a motor that backfired then it would not start, turned out that when it backfired it sheared the flywheel key and through the timing off. New key and we were good to go. Maybe also a bad crank seal.

Did you remove the high speed jet when you cleaned the carb? Fuel might be flowing into the carburetor bowl, but might not be getting through the high speed jet.

OK, you had the carb apart and you removed the high speed jet, all is clean.
The motor won’t run unless you prime the cylinders with fuel, then only for a few seconds.
Fuel runs from the carb when you remove the bowl drain plug and pump the fuel line, correct?
How about the carb throttle linkage, perhaps the butterfly is stuck open somewhat. I’m pretty sure the choke linkage on that engine holds the butterfly open a bit to help start a cold engine. So, if the butterfly is stuck, or the choke linkage is messed up, this might explain the no start. The engine starts when primed, but then cuts out/spits/backfires because the throttle linkage is stuck open somewhat.
I’m assuming no major work was done to the engine (other than the carb) since the engine was last run many years ago, no one has messed around with the intake manifold have they?
You might try opening up the low speed needle another turn just to help get it started and keep running, then you can dial it in once the engine will keep running for you.