OK, thanks for the clarification about blowing air through the carb. Sounds like everything is OK here. With the carb assembled, air passes only when the diaphragm is pushed in. Good. Earlier I suggested to make sure that the choke is fully closed. Most choke butterflies have a hole drilled in them to allow some air to pass. So all I mean is to make sure the choke is as fully closed as it is designed to be. If you use a choke servo, it could be adjusted so that it is almost closed but still cracked open a little. If you use your finger to close the choke, usually, they snap into the fully closed position pretty easily.

One of my buddies had a bunch of "lint" on the carb inlet screen. (this was a zdz 80). It was hard to start (50 flips required) , but somewhat surprisingly would run reasonably well once started. I think you have something similar going on. Is the plumbing from the tank to the carb confirmed to be free flowing? No valves, filters, etc? How about air leaks in the crankcase or carburetor fitting? Air leaks will cause poor fuel draw.

Again, my experience with my Tillotson carb is that every flip of the engine (with throttle cracked open and choke closed) will bring up a few inches of fuel in the tubing. After 4 or 5 flips the engines fires and quits (because of the choke being closed). Sounds like in your setup, there is very little fuel being drawn on each flip of the prop. Can you see this happening?