Sometimes, letting professionals help is the only way. They already know the places that can accommodate your party's size, and can often get very competitive rates.
Similar to apartment locators, the "agent" is often paid by a commission from the apartment rather than by the person using the service. So this is one of those cases where you are the product.
No doubt. But probably not specifically for this stay, so basically all the people who also rent this place without using a travel agent are subsidizing your use of the travel agent.