Spark... air... fuel.
Sounds like you checked spark, checked air - at least seems like it's pumping enough to work by your hand test...
...so how about fuel?
I'm new to carbs so I'll have to defer to someone with expertise there, but I'd check some fundamental things -
Besides making sure there's [good] gas in the tank, I'd disconnect the fuel line just to make sure gas is reaching the carb, and my quick "is it the carb?" test would be to spray starter fluid straight into the carb and see if it at least starts and runs for a second or two. If it does, that would seem to isolate the issue to something in that mysterious can called a carburetor.