Since you cut the pump out, put it back in with unions and a real brass or SS check valve, not plastic from home depression stores. Some of those plastic ballvalves are surprisingly durable, but if you are re-doing the whole set up, go for a brass one. Apollo is a good brand if available.
I really think you need a valve on that "priming" tee. Then you can prove air or water easily as the pump tries to make pressure, and use it to release air. If you keep getting air out of the valve, you have suction side leak. How can you have this setup without a pressure gauge? Should have always been one at the tank outlet and the pressure switch. I would have said it's time for a pro, but since you were able to get the pump apart, you must be pretty sharp. There are simply some nuances to pump work that take a lot of time to learn. But you are in a good spot to pick those up right here!