By any chance, did you mistakenly put a fuel filter that's for engines with a fuel pump? B&S engines without a fuel pump seem to require the small filters with a screen in them. I'm assuming the carb doesn't have enough power to draw from the paper filter.
Yes, this assembly has a filter between the tank and carburetor.
1) Check out salcuta's remark. Sometimes over-engineering these lawnmower engines is not a good thing.
2) qman has excellent points. First eliminate the fuel tank as the problem. Does gas flow out of the bottom of the tank?
3) If gas flows well, your problem is the carb's float bowl. If you can't start it again until you press the red finger primer pump 3-5 times to get FORCE more fuel into the carb bowl, then the carb bowl isn't allowing gravity to feed the bowl. Why? Because the pin is dirty and does not recognize that the bowl has run out of fuel. The float bowl may not even be moving!
4) Ghetto Solution #1 is to remove the float bowl (one small nut), and spray the heck out of the insides of the carb bowl with carb cleaner. Wait an hour, then spray again. You can even re-install the float bowl and fill up not with gas, but with Seafoam or liquid carb cleaner and let sit overnight.
5) If the ghetto solution does not work, then yes, remove the carb and rebuild. donyboy73 has some EXCELLENT YouTube videos on how to do this. You will need a carb re-build kit (under $15) and that will come with new gaskets including the one for the float bowl AND the one that mounts the carb to the B&S engine.
If a lot of gas (a full tank) keeps the engine running- that is because the weight of all that extra gas is pushing fuel into the carb. If at 1/4 tank your engine dies every five minutes, your float bowl is dirty- meaning that needle is dirty, the spring is dirty, the small holes are plugged up, etc.....it all needs to be cleaned.
I have a 13 year old Toro with a Tecumseh engine- it starts on the first half of the first pull every single time. I ONLY run 100% gasoline, no ethanol at all. The gas can sit in the carb for a couple of weeks in summer and it does not matter. Ethanol will kill your carb- even if you run Stabil. For Stabil to work properly, you must add it to the gas can same day as you filled that gas can, AND, the gas from the station can't be older than 3-5 days. Stabil added to 7 day-old gasoline won't save you- it will gum up the carb.
This is why I advocate for the beating and deportation of everyone connected to the ethanol industry.
1. Addressed, see above
2. I'll check that out, it's a great point
3. This engine does not have a primer bulb
4. This is a new carburetor, see my OP. Original carburetor was rebuilt for a different fuel delivery issue (but this fuel delivery issue was already present). I had hoped that rebuilding the carburetor would resolve both issues, but it didn't. It only solved the other one. Later replacing the original carburetor, this issue persisted.
5. No need, brand new carburetor
Yes, a full tank pushes fuel into the carburetor so when that depletes, the engine dies (fuel, spark, air). My lawnmower dies with the tank at 3/4 full!!!
I've never needed to run Stabil, on any lawnmower I've ever used growing up when I mowed the lawn at my parents' house, so I never bothered to use some on mine now that I have my own lawnmower, but will consider it.