[AZKen]

OK, will wait for results of those tests before suggesting cause and next steps.
As we know, a motor needs fuel/air, compression and spark at the right time. So with a "no fire" situation we troubleshoot those. Your motor starts with gas or fluid sprayed into carb throat, so we can concentrate on fuel delivery. When helping from afar, we only have the words of the post. We"trust" the description as much as possible. You have been very specific about the history and present issue. That's good. So using your words, logic and experience, we give ideas.

It seems the carb bowl is empty after sitting due to evaporation.. There should be enough for the accelerator pump to give a squirt or two.
Somehow or another there is not enough. THIS CAN BE NORMAL and just takes a special method for starting. We have all had motors that sit. You say a few days. That can mean 2 days or 5 days. You will just need to set to full choke, pump pedal like crazy til it starts. Then gradually open choke.

You tried an electric fuel pump, which if was installed correctly and was the correct PSI...should have started the truck. Absolutely. So this throws me off. Something fishy there. This causes me to think of spark and time, but it starts and runs good....so I bounce back. When you have an electric fuel pump, which I really don't recommend, you have instant pressure. You don't need motor to crank to fill the bowl. Bowl fills and accelerator pump (if working) gives a gas shot. Like the squirt bottle or ether.

Fuel delivery problems have many causes. Empty gas tank, tank not venting, fuel lines or tank clogged, fuel filter clogged, fuel pump, sticky needle valve, float setting.....all that stuff. But again, and I'm sure you are thinking the same thing...it starts with a squirt and starts fast and has new pump and rebuilt carb.

So conclusion:
Most likely,
1. It's normal and is sitting longer than 2 days. Bowl empty. Needs violent start procedure.
2. It's the accelerator pump.


Other, including items in green above,
3. Bad new pump
4. Carb rebuild problem or old rebuild....or Ethanol has attacked needle rubber and accelertor pump rubber. Your careful test, after truck sitting, to observe squirt before trying to start will help to determine carb issues. This will determine if bowl has gas after sitting a few days. Report how many days truck sat when test was done. Also you say the seller used gas to squirt so that is good to know. Kind of weird he needed to do that to start motor, then you put on a new pump and rebuilt carb, among other tune-up items, and it does exactly the same thing. What am I missing there? I think there is a hint there somewhere. Did you ever empty and clean the tank?
5. Choke not fully closed
6. We may want to do a pump flow test at some point.


I should have said check for vacuum leaks. Hoses, gaskets, open ports not noticed on carb or on intake manifold, cracks. You say one hose. That would be vacuum advance?

Theory:
Clogging somewhere causes slow fuel pump flow to fill an empty bowl at starter cranking speed. When artificial starting occurs with squirt in carb, motor runs much faster, pump overcomes clog to fill bowl and stay running except during heavy acceleration. Something to think about. Except electric pump didn't work either.