If this were a site that I wanted to make tons and tons of money, that would be one thing, but really I just want it to be useful. Brand isn't so important and so if people see a Google logo on something, I don't think it's the end of the world. And if the functionality were a necessity, I'd definitely never consider using something I didn't have any control over to solve the problem. However, we really need very little document processing functionality. Mainly I want to offer people an easy way to copy and paste.
If I mistrusted Google, as I know a lot of people do, I'd also probably be quicker to consider reinventing the wheel. But I don't. Google produces useful tools and products, doesn't misrepresent their intentions or stoop to gimmicks, and is a company I think of as fairly ethical. That could change somewhere down the line, but if it does, it's easy to yank references to an external API and replace it with something written in-house. In the short term, my goal is just to get something working. That said, though, I'd never consider using Google for site search. So why is that too cheesy, while using Google Docs isn't?