Cant start B&S motor on old Rover mulcher

I agree with SRTsFZ6. It sounds like, when you said you cleaned the carburetor, you only cleaned the outside, not the jets, nor the passages. The picture you posted looks like one of those B&S units that sits on top of the fuel tank. It has no fuel filter other than a fine screen at the bottom of the intake tube. Since the fuel is not gravity fed, the carburetor has to pump the fuel up to itself. It's common for there to be a long tube leading down to the bottom of the tank and a shorter tube leading to a small well. The carburetor pumps a little squirt from the tank via the long tube and dumps it via the short tube into the little well in the body of the tank just below the gasket. The carburetor, through normal engine vacuum, then sucks fuel up from the little well through the main jet into the throat of the carburetor.
 
If I guessed right, we can identify a few things that can go wrong:
 
1. The screen at the bottom of the long tube has a lot of junk stuck on it, the result being that some fuel gets through, but not a lot. The engine would probably start, since it's pulling fuel from the little well, but quickly run out because the well is not being replenished. So check that the screen is clean. If it's dirty, also check the tank since the garbage had to have come from somewhere.
 
2. The little well is full of junk, the result being that the main jet cannot suck fuel from the well into the throat. Check that the well is clean.
 
3. The main jet is clogged. If there's junk in the little well, some of it probably got sucked up through the jet and clogged it. Flip the carburetor over and shoot some carburetor cleaner into the jet while you're looking down the throat of the carburetor. You should see cleaner shoot through the main jet passage into the throat.
 
4. The pump diaphragm is stiff or cracked. This is what SRTsFZ6 is talking about. The way that the carburetor pulls fuel from the tank up into the little well is with this ingenious little pumping mechanism. When you tear the unit down, you'll see a black gasket with a bunch of holes and a somewhat loose area (the diaphragm), about the size of a quarter, with a metal stem through it. The stem pulls and pushes the loose area, creating suction to pull fuel from the tank, and then because there are flaps functioning as one-way valves, pushes fuel into the well. With age, the diaphragm looses flexibility so it doesn't move up and down with the metal stem so well. It can also have a rip in it, in which case it will never develop suction or pressure, and so do nothing. Shine a light through it and look for holes or tears.
 
My guess for why the engine initially started is that the choke isn't only a choke (which more or less blocks the air intake) but also a primer (which shoots a little fuel into the throat), bypassing the pump diaphragm and any trouble the diaphragm might have. But after that little fuel is used up, the engine now depends upon the failing pump diaphragm, and so it runs out of fuel and stops.
 
Maybe I'm guessing wrong. Maybe I'm even envisioning a different carburetor. When you next lift the carburetor, send us a few photos of the top of the tank, and of the bottom of the carburetor, and we can be a little more sure of what we're saying.