It's not that you wouldn't be able to bind wires -- it's that you'd be forced to _explicitly show_ the binding together of wires with \$$\otimes\$$, before it ever reaches a reaction node. Right now we're able to define reaction nodes as taking some number of wires in and putting some number of wires out -- but if you look closely at how \$$\le\$$ works, reactions \$$x \le y\$$ relate _precisely two objects_, \$$x\$$ and \$$y\$$. It's only because of associativity that we can unambiguously represent a product object by the ordered collection of its constituents, and thus write a reaction node as operating over an ordered collection of wires.

Without associativity, we would require an extra kind of node, specifically for \$$\otimes\$$, to either bind together two wires into one or split apart one wire into two, so that we could explicitly describe how wires are bundled together before connecting the single combined wire to a reaction node.