I have a Briggs & Stratton 20hp Vanguard model 351777 type 1036-E1 motor, recently started running very rich lots of black smoke and poor running. Stripped carb but could find nothing wrong, float and seat sound, shuts of fuel very well. Motor runs fine without air filter element,put element on and it struggles and stutters (you can smell the un-burnt fuel) whipping the element of again the engine recovers and runs fine but lots of pooled fuel visible in mouth of carb. My problem is the air filter is a brand new B&S unit. The old dirty filter causes the same problem. Interesting twist... I have a 2nd mower same engine, used filter from this machine on problem machine works fine (grass and crud on the sponge cover included) however the used and new filters from the problem machine cause exactly the same symptoms on the 2nd machine (flooding). I can't get my head around why one dirty filter is Ok and one isn't and why the brand new one is also a problem. Have I missed anything??

Little bit of dishwashing soap and a few minutes at a sink and they would both be clean...

There is something you are overlooking somehow. Something blocking airflow.