If it starts and runs for a few seconds with a prime, then the carburetor is still not metering fuel.
When you cleaned the carburetor, did you remove the main jet and nozzle and inspect the orifices to make sure they were all clear?
Carburetors for GCV Honda engines are cheap. You can save yourself alot of grief and just install a new carburetor and be done with it.
Ya gotta know how it WORKS
, before you can figure out why it WONT