Coinbase is not an exchange, they're a retail broker, the fees are way too high for trading.

As mentioned, Coinbase is not an exchange and honestly leaves much to be desired (limited instant buys, for example). I think the better reason is that allowing any single Bitcoin trading company to get too big is unhealthy (see MtGox). Having competing companies is very much in the interest of the btc community.

Not really an exchange, is it? More of a payment processor? Could be wrong.

No you're right. Coinbase does payment processing and hosted wallets. Users can buy or sell bitcoins to Coinbase, but it's not an exchange in the sense that any two users can trade USD/BTC with each other.