It is well-known that any symmetric monoidal category is equivalent to a strict symmetric monoidal category. The construction of this strict monoidal category is rather technical and it appears to me that there is a significantly easier way of obtaining an even better result. Of course this means almost certainly that I am making a mistake.
Given a symmetric monoidal category $S$, consider the category of isomorphism classes $T$. Explicitly, let $T$ be the full subcategory of $S$ which has one distinct object for every isomorphism class of $S$. The inclusion functor $F:T\to S$ is fully faithful and essentially surjective. Hence it is an equivalence of categories with some inverse $G$.
Now $T$ is strict symmetric monidal with operation $\oplus_T$ defined via
a) very simple
b) also commutativity is strict which appears to be better than the classical result.
So what is going wrong?
Edit: This is basically what André is saying in his answer. The skeletal category is certainly strict with a different associator. For some reason I thought that one had an additional natural transformation somewhere so that the associators don't have to be compatible in a strong sense. So we get an equivalence from a strict monoidal category to a given category in this way, but the functor is then not monoidal.