The symptoms described is a signature issue on this type Briggs 6.5 Carb (and others that use it)
If any debris gets in the fuel and fits thru the float valve it settles at the high speed jet that is at the bottom of the fuel bowl. In fact it is the brass piece that holds on the fuel bowl to the carb.
I find it best if it comes down to having to fiddle with this jet, to remove the carb, get some Lacquer thinner and a small brush and thin wire and really clean clean clean all dirt off the outside. Then throw that Lacquer thinner away, clean the container and take the hi speed jet screw out of the fuel bowl. If you are careful and gently pull things apart, hopefully you can save the rubber gasket between the bowl and carb. If it sticks, wd40 it with spray and keep working it off.
Next, Ya gotta have some air pressure to blow out the carb, and you first should let it soak for a while. Take the float off and shake it to make sure there is no fuel in it. If it is leaking and I have never found a plastic float that leaks, it will have to be replaced.
Once the carb has soaked, wire out all the holes and then soak again and wire again, and then blow out every hole you see. Make sure to put the air tight against the float valve hole.
Reassemble, the reverse of taking it apart, reinstall, and next, Clean the filter with air and clean the filter holder with the left over carb Lacquer thinner, so that you are starting out clean.
And don't take any short cuts trying to prime and start the engine without reassembling everything. All the parts including the filter are required to be installed to get the silly thing to run.
Prime it with the prime bulb and pull away!