It shows you're serious. Most people complain to blow off steam. This is what customer service handles. If you want something done at the corporate policy level, e.g. to have policy changed or have a decision made per policy reversed, you attack at higher levels.

The traditional form of leverage is legal. Have a lawyer pen a letter gets you out of customer service. It is as effective as it is expensive. Next best is a regulator (you can think of these as narrowly-scoped, publicly-funded lawyers). Unfortunately, nobody regulates Google.

So your final threat is getting a lawmaker pissed off with you. This is less about passing legislation (it's hard to pass legislation; everyone knows that) than creating an official, reliably-corroborated paper trail which could go public, causing PR damage, and/or damage relationships the company may want to lean on in the future.

More practically, I don't want to sit around writing and responding to letters. Having someone else do the back and forth with me Cc'd is more pleasant.

Disclaimer: I am not a lawyer. This is not legal advice. If you need legal advice, contact a lawyer.