For example, associativity is inherent in the notation: A→B→C→D is the only way to express the composition of three functions, you can't even write the difference between (f∘g)∘h and f∘(g∘h).

It's only unwritable in a linear picture. If you wanted to make associativity visible, then you could express it in terms of commutativity using arrows that hop over nodes.