The mechanical pump doesn't care if it has fuel to pump or not for a couple of minutes. It's going to do what it's going to do. If it's a priming issue, there's a couple of things you can do that is minimially invasive.
If you know the fuel suction line is clear, then take a plastic syringe (without a needle) and suck up a little gas and fill your primary carb bowl with gas through the primary bowl vent tube on the carb. Believe me, it works. I have done it. Don't go crazy, you just need enough to fill the bowl part way to get the engine going. You don't want to flood it. On a holley, you can remove the bowl side float level screws for when setting the floats, put enough fuel in to fill it to the screw holes to get an idea how much to fill. Then you can put the screws back in. You may need to do this a couple few times but it will allow the engine to start normally and run on its own until it pulls fuel into the line by itself. It WILL do it if the line is clear and fuel is in the tank.
If you suspect there may be some pluggage in the line back to the tank, you can, as mentioned, pull the fuel line from the suction side of the pump. Is the hose wet with fuel or fuel draining out? If so, it's probably getting fuel and the problem might be in the pump itself. If not, you can do a couple of things. Assuming you have a 2 tube sender on a stock tank and an air compressor, you can remove the fuel tank cap, use VERY LIGHT air pressure on the disconnected suction line at the pump and puff air back through the fuel line. Have someone back at the tank listen for bubbling. If you hear bubbling and/or venting of air from the neck, you have a clear line. Then, reverse the procedure to prime the line. Again, light and slow with the pressure. You won't need much. You don't want to blow off the fuel sock or pop a hose by going crazy with air pressure. Cap off the canister vent line from the tank, and slowly and lightly pressurize the tank from the neck side and have someone with a catch can at the suction line of the pump. When fuel pours out, stop and reconnect line to pump. Check under the car to ensure there are no other leaks. Fix any if there.
Good luck.