I have the same 6.5 b&s engine that used to smoke like crazy while cutting my lawn, I pulled the mower from a trash pile and all it needed was a good cleaning and it ran.
I never changed the oil in the past 6 years since I got it.
It always smoked and needed oil every time I cut the lawn.
so over the years I added oil and it smoked, I thought pull the engine find the source, but it kept running and cutting the lawn.
why fix something that works? especially if it cranks up on the first pull every time.
over the years the smoking went away and so far this year it hasn't needed a single drop of oil.
I just cut my lawn today, starts on a single pull and uses half the fuel it used to use 5 years ago, I guess the engine is happy with the Amsoil I have been adding to it all this time, AME 15w40, after all this time the oil is still clean on the dipstick.