If a problem can be worked around by varying the throttle setting then the only cause is lack of fuel.
So some where from the tank outlet to the main jet inlet there is a fuel blockage.
If you want to rule out the shut off solenoid then replace it with a bolt.
Oddly ever we call them bowl nuts but I have never seen a bowl secured with a nut.
Pinch a fuel tank off something else and rig up a temp fuel supply then hook it in at the carb, and at the fuel pump to try & identify where the blockage is.