It does sound as if rings are the culprit, but there is a simple test you can do that will usually verify it. Do a compression test, put a spoonful of engine oil into the cylinder through the spark plug hole, and repeat the compression test. If the compression increases markedly, the compression rings are leaking. With Briggs engines especially, if the compression rings leak, chances are it will burn oil.
It is possible to have vast oil consumption and oil smoke you can't even see through, together with good compression and no oil going through the breather (happened to me yesterday with a Honda engine, it actually surprised me, it's been so long since I last saw it) but so far at least, I haven't seen that with a Briggs engine. If the oil test says the compression rings leak, you need new rings anyway, but you can be pretty confident it will fix your oil smoke as well. If the compression rings aren't leaking, it is still probably an oil ring problem, but please come back and tell us so we can talk about it some more.
