Vote #1 Michael Minnich: Michael Minnich’s answer to Linguistically speaking, why is the relationship between the signifier and signified mostly arbitrary? It brings up several pertinent reasons.
My answer’s simpler: restricting ourselves to lexicon, non-arbitrary signifier–signified relations in a spoken language are going to be limited to referents that make a sound. Most verbs and adjectives and abstract nouns defy onomatopoeia. Even with concrete nouns, what’s the onomatopoeia for “hair”? “sun”? “fish”?
Even if we broaden things to include indexical signs and sound symbolism, there are real limits to how much can be signified by a non-arbitrtary signifier in a sound-based language. The limits will be enough to prevent you having a real communicative language—even if you don’t move up from lexicon to syntax.