If I correctly understand, you are misinterpreting the meaning of the product and sum of observables.
When you say "We can now define a sum and a product of observables. These are obtained by performing the two measures and then adding or multiplying their values."
This cannot possibly describe the usual sum A+B and product AB of operators. For the product, it is not even hermitian unless A and B commute. Agreed, A+B is hermitian, but the spectrum of A+B does not contain the result of the sum of a measurement of A followed by a measurement of B (in either way), again unless A and B commute. For a counter-example take A=[[1 0];[0 -1]] and B=[[0 1];[1 0]].
I hope I correctly understood your question.