There are 3 basic carb designs that I know of: 1) float (characterized by a bowl and usually a spring loaded valve to drain the bowl), 2) Diaphram and 3) Direct feed where the carb sits on the gas tank.
Gum and varnish accumulate, especially with fuel left in the engine/tank for a long tiime untreated with additives, like Stabil. I treat fall/winter gas for both 2 cycle leaf blowers and 4 cycle snow blowers. The gas could sit from November to May. Not good.
Well, the gum acts as glue to the fuel valve and varnish restricts passages in the mixture needle valves or makes the fuel valve not seat properly.
In the float type carb, the fuel is gravity or pump fed and stopped by a small valve in the carb. That valve basically lets fuel into the bowl and stops when the bowl is full, because the valve is actuated by a lever arm of a float. If the float gets a hole in it, the carb floods. If the valve sticks, no fuel. Debris in the fuel system may also cause the valve to stay open.
In the diaphram type carb, the valve is very light and small. Gum can easily make it stick.
Engine vacuum basically activates the diaphram and the diaphram is connected to a very small fuel valve. A miniature pump so to speak. If the diaphram gets a hole in it, no fuel.
If the fuel valve sticks. No fuel. If there is debris in the gas, the valve may stay open and the carb floods. The diaphram carb sometimes has a primer bulb attached to it.
Design 3 is not likely in your case.
You already confirmed it's not the gas shut off, fuel filter or fuel pump if equipped.
Some carbs have two mixture screws. One for the idle RPM range and another for high speed.
Both
Welcome to the Parts Store and
briggsamdstratton.com have engine diagrams on thier website.
The only real cure is to rebuild/clean the carb. This requires disassembly.
I'm not sure of the procedure to set the mixture on the newer engines, but in the old ones, you'd just bottom out the mixture screws and open about 2 turns. Start the engine and adjust until it sounds right.
Cars, on the other hand, in later years cars required drilling out the caps on the mixure screws. I set the idle/mixture using propane gas and a tach which would get the emissions right.