You should be able to push air through the pick up tube. Soak the carb tube with carb cleaner or have the carb ultrasonicly cleaned. Blow through the tube with compressed air or bottled air carefully. It should open up. Reinstall carb with a new diaphragm and gasket, make sure you install them correctly. These carbs are so simple I don't see a reason to replace.
How old is this engine? If the primer bulb is old and not flexible it will stick and not return to it's proper position. If the gasket and diaphragm are installed in the wrong order nothing will work. You need todo some checking. When you report back, please include model, type and code numbers.
If you would have posted the code number, i would be able to tell you the engine age. When you assemble the carb to the tank, diaphragm rests on the tank, gasket will go against the carb. Fuel pump spring rests on the diaphragm, do you have this spring? When tightening the screws the throttle shaft must be held in the wide open position. There is no vacuum when pushing the primer bulb, the primer puts pressure on the diaphragm, and working with the fuel pump spring, pushes fuel into the carb throat. If not assembled correctly or missing the pump spring, will result in your condition. Have you got a Briggs L-head service manual to help you. If not they are available free in PDF form on the Internet. At the present time we are unable to posts links here, so you will have to do a google search.
Does it run with the air filter on? Your choking it to keep it running which would most likely be a fuel restriction, may also have an air leak.