This is completely legitimate, right? There must be a most advanced civilization. Why can't we be it?

Yes, that's possible. It just goes against the Copernican Principle (http://en.wikipedia.org/wiki/Copernican_principle), and that's why many people are not happy with that explanation.

Well... Someone always wins the lottery.

You can also twist it around: if we were the first ones, what would we see?

One can also imagine the window between the dawn of technology and becoming undetectable is short enough the odds of finding a peer are very small.

There's also Anthropic Principle (http://en.wikipedia.org/wiki/Anthropic_principle) which states pretty much the opposite to what the Copernican says.

So a more advanced civilization would have a better spot in the Universe? I don't get your point.