Most stats providers compute unique visitors using client-side cookies; not entirely accurate, but quite cheap. You're right though, it's a lot easier to use an existing (and free or nearly free) tool than to build your own.
That, and the fact that I prefer to process my logs on my computers. As a matter of principle (decentralization, privacy, freedom, blah blah).
Now, I barely know Analytics. But it looks cool, so I'll check, and see if I can reap most of their benefits myself. (If I can't, I'll have a tough choice indeed.)
(Something about that bar not being updated until the first byte of the stream was read -- so if a static.foo.com was hanging connections, it would get stuck blaming whatever loaded right before it.)
GA now has asynchronous tracking though, which solves this pretty nicely. Though I'm not a JS guy, my understanding of it is that instead of sourcing the script and then issuing the tracking calls, you put some instructions into a global variable, and then source the script. Nothing blocks on the script's loading, so even if GA is down the rest of the page is supposed to load correctly. Once the GA script loads, it looks at the global and runs the tracking commands.
But still, I would have liked the analysis process to be completely invisible from the client.