"One-way" fuel line valve..?

Pumping squirts gas, which sounds like you need to start cold.
You probably need a new pump.

Maybe, my pump is original and 53 years old. But it delivers 5 psi at idle. So, I don't know if it's shot.

Yes, The carb accelerator pump gives a full squirt of fuel that way.

I will try to start with the gas pedal fully depressed. Thanks