The easiest way to check the condition of the rings on a Briggs is to do compression tests with and without a teaspoon of engine oil put in through the spark plug hole immediately before you do the test. If the pressure is 15 psi or more higher with oil in there, you need new rings.
Oil in the breather tube is often an indicator of worn rings, but it is an even better indicator of an over-filled sump.
Very worn rings, to the point where the engine smogs out the neighbourhood, result in liquid oil in the combustion chamber, which you do not seem to have. Beyond that I can't tell: black deposits are normal, and in the pictures they do not seem heavily caked up. How much blue smoke was it blowing?
