Oh no, you don't have that carb?!
The engine numbers I was talking about are those on the shroud just above the spark plug and throttle cable.
Mind you, I claim to be nothing more than an amateur that has had some luck at getting engines to run. I am, however, a professional at taking the satisfaction of hearing an engine run that previously hadn't. There is no doubt more folks out there who know much more than I about such matters, but if nothing else, I'll give you my 2 cents worth, and lots of moral support.
Anyway, backing up, here's my check list.
All the hoses, gasket, and rubbery bendy things are in good shape, check.
All the cables were put back into the same holes they came out of on all the various pivot arms, check.
Now that I think I'm done, no extra parts left on the work bench, check.
All of the controls move and pivot freely as they should, check.
It sputters and acts like it wants to run when you pour gas into the spark plug hole, check.
By putting the gas in the spark plug hole, you're basically choking it. I acts like it's going to run, but stops when that bit of gas you pour in is gone. So, it's not pulling the gas from the tank through all the tiny little holes the carb and delivering it to the cylinder where it can do what all good engines should do.
Does the gas run readily out of the gas tank when you pull the fuel line from the carb? If not, the clog is somewhere in the fuel line or in the tank.
Does the gas run freely out of the bottom of the carb when you remove the nut and bowl. With the bowl out of the way, can you stop and start the flow of fuel by raising and lowering the float. If not, there's a clog in the float valve.
If the gas is spilling about and making your hands all smelly, and dirtying up your garage floor to this point, then there must be a clog between the bowl and venturi keeping it from arriving at the final resting place for combustibles in the engine.
Take a look at these photos and directions to see if they look anything like your carb. Maybe this will help.
Disassembly of Briggs and Stratton / Walbro LMS Carb on Briggs Engine 125K02 0137