Dan: here's another way to answer your question, which may be more to your taste. You suggest that

- \$$+_1\$$ is a way of combining elements from \$$S\$$ (?)
- \$$+_2 : \mathbb{N}[S] \times \mathbb{N}[S] \rightarrow \mathbb{N}[S]\$$ is the monoidal operation.

However, you don't say what the first operation maps from and more importantly what it maps _to_. If you try to figure this out, you will probably be led to a framework where all you care about is \$$+_2\$$. Give it a try! And notice that we can (and should) think of \$$S\$$ as a subset of \$$\mathbb{N}[S]\$$. So, there's ultimately no point in having a separate operation that only lets you combine elements of \$$S\$$.