It's hard to diagnose your problem but, it's more than just a primer issue. However, I'll try to explain how the primer works. There should be a line that comes out of the fuel tank to the primer and another line from the primer to the carb. The function of the primer is to take raw fuel from the fuel tank and dump it into the carb venturi to allow for a rich mixture of fuel to get into the engine. It sounds like your primer is in working order if it's pushing air on the carb side however, from what you say, it sounds like it's not pulling fuel out of the tank. You'll have to ensure all connections are tight, lines clear and the fuel pick up end in the fuel.
With the carb assembled and installed on the machine and it fires with starter fluid, that tells me that the carb's main jet might be blocked. As previously mentioned, you'll need to pull the main jet out of the carb and clean it. Also, the bolt that holds the float bowl on might have a couple of small holes drilled in it across the diameter of the bolt. Clean them too. Try to clean as many jets and orifices you can see with a very thin wire, WD-40 or carb cleaner and blow out with compressed air if you have it. Let us know how it goes.
With the carb assembled and installed on the machine and it fires with starter fluid, that tells me that the carb's main jet might be blocked. As previously mentioned, you'll need to pull the main jet out of the carb and clean it. Also, the bolt that holds the float bowl on might have a couple of small holes drilled in it across the diameter of the bolt. Clean them too. Try to clean as many jets and orifices you can see with a very thin wire, WD-40 or carb cleaner and blow out with compressed air if you have it. Let us know how it goes.