I have been having this EXACT problem! I didn’t figure it out until I tried to gently blow out the fuel line: I heard a quiet “pop”, and 5 gallons of gas dumped out onto the ground! Well, I caught most of it in a bucket.
It turns out that, though the fuel lines looked fine, the inner layer had turned to gook and separated from the fabric and outer layer. Presumably from the ethanol gasoline.
It seems like once it was running and a flow was started, it could pull enough fuel through to run. But when it had been sitting, the suction of the fuel pump on the softened, delaminated hose made it do what soft suction hoses always do: collapse and block flow.
The “pop” I heard, and what leaked all the gas in a matter of seconds, is that the grommets in the bottom of the fuel tank - that hold the petcocks - has also dissolved in the ethanol; that’s what blew out when I put a little pressure on the system.
Cheap, simple solution to something that’s been driving me nuts for weeks. Lessons: don’t replace a fuel pump and rebuild a carburetor before you replace $5 worth of fuel lines. And if they have E10 in your state, seek out a source of non-ethanol “off road” gasoline!