OK! You know if you put fuel in it runs. You have replaced the fuel lines and filter which is great too. What is left? Two things....
1) fuel inlet screen inside the carb.
2) diapragm lever arm clearance.
Both of these are under the little covers on the carb. The one with four screws is the diapragm. The other will have one or two screws which is the inlet screen side.
Remove the gas from the vac and proceed to take the fuel inlet cover off.
The cover is under the air filter body so you have to pull the two star head screws out to get to it. With the fuel inlet cover removed you can pick out the screen inside the round hole and check it for debris/gelled fuel. These things plug at the drop of a hat! You can either try to clean the screen or go to a parts store and pickup a fresh one. I would go with a new one since the screens are so delicate cleaning is almost impossible. If you hold the screen up and look through it you will understand what I mean. You can also look on Ebay and order one and get it by mail. Which might be easier and still cost effective.
Once you replace the screen *(you put in the new by setting it in the hole and gently pressing it in with something blunt and about the size of the hole. It wont take much effort so be gentle. Once the screen is in put it back together, add a little fuel, and check the primer for rebound.
That is to say that the primer , once pushed in, should come back out quickly and fill with fuel. If it drags or stays in you still have a block somewhere in the system. This might mean removing the carb and stripping it completely and cleaning all the passages.
Do the screen thing first and we can discuss the diapragm work after that. Here is a link to homelite too: