It is a 217802-4145-F1 for those don't have access to the serial number decoder.
It is because the solenoid needs to be powered when ever starting or running. And the fuel bowl without the solenoid was available at one time but Briggs no longer sells it. I was lucky enough to get three of them before were all sold out.
So try powering the solenoid directly to see that helps for troubleshooting purposes. Also you can remove the solenoid and install a M8-1.00 screw temporary.