Doesn't the carburetor bowl have sufficient fuel in it to start and run the rig for at least a minute.
As soon as the engine started, the system would fill and no one would be the wiser, especially with a electric pump...as soon as the key is turned on, the pump starts.
On the clear inline filters (throw-away), I have noticed that the fuel never does completely fill them, even when the engine is running. As Baldy pointed out, whatever type pump you have, it is equipped with check valves of some sort.
I think if the rig won't start, it is a carburetor issue, and has little to do with a system that might be draining.
Stuart