I recently had similar symptoms on my Case 570. After sitting overnite, it would start then die. Keeping the rpm's up would help at first but then it started dieing when running. Found a replacement mechanical fuel pump on ebay super cheap and it would not prime when working the plunger. Neighbor started checking fuel lines for blockages and found one of the three lines from the sending unit to the tank was blocked. put everything back together and the pump still would not prime. Neighbor suspected my super cheap ebay pump might be faulty so he put the old one back on. Primed quickly and no more problem. Sounds like you are close to finding your problem. Good luck.