The point you clearly missed is that even if the universe was teeming with SPACEFARING civilizations, we would be extremely unlikely to know about it even after ten centuries of searching the sky with our telescopes. We've only in the last 20 years even confirmed the existence of extrasolar planets, and most of those only by their gravitational effects on their parent star; even at FTL velocities, something as small a a spacecraft would be WELL below our detection threshhold unless it was relatively close to Earth orbit.
In other words, they would have to go out of their way to visit us before we ever knew they were there. That in turn implies they would have to have a REASON to visit us, which -- if they don't know about us -- they would not.
Exactly right. The amount of time we've been creating transmissions is extremely short. Just 50 years ago, we were preparing to send the first man into orbit. Our "footprint" in the cosmos is next to nothing. We are situated on the periphery of the Milky Way Galaxy. There's no plausible reason for anyone to travel a vast distance to our solar system, unless they sent probes out many hundreds or thousands of years ago and found our planet habitable enough for their purposes to come for visitation and/or colonization some time in the future.
Our imaginations are adept at creating ideas that go way outside the boundaries of plausibility. The biggest hurdle to face is the energy problem. The energy required to get to FTL and then slow down to a stop is immense. Much greater than we could ever hope to see accommodated in our lifetimes and even the next several hundred. But even if solved, the distance to travel here, let's say from near the core of our galaxy, is way too vast for any alien race to bother. Even if a civilization developed a wormhole technology, some way to fold space so distance would not be a problem, there's still the matter of a very sparsely populated region of solar systems with planets worth visiting.
Just as primitive human beings thought the universe revolved around the Earth, so do many people today still believe we have a jewel of a planet that any alien civilization should want to come visit, and that they can. The sad truth is that we're too insignificant and far away to be worth visiting. In all likelihood, the human race will eventually die out someday, alone in the icy cold of space. The only thing we will probably achieve is to record bits and pieces of our lives and send them off on a probe into deep space in hopes that someday it'll be discovered and our presence in the universe will be remembered.