> Imagine someone saying, "We're doomed because people walk very slowly
I think I may not have made myself clear. The analogy here is not that "we're doomed because people walk slowly". There is nothing that happens here on earth that is even remotely analogous to the constraints that the speed of light imposes on interstellar processes. Even at a leisurely walking pace, a human can circumnavigate the planet in less time than it takes light to reach the nearest star. Human walking speed is blindingly fast (relative to the size of the planet) compared to the speed of light (relative to the distances between stars).
The reason we're doomed is not that we're not walking fast enough, it's because we won't stop walking and take time to think about where we should be going. We just go.
But my point is, "we just go" is exactly how we conquered the Earth. We didn't think, "let's colonize the whole planet!", we just went. We walked from Africa all the way to the tip of South America.
We weren't doomed then and we're not doomed now. We will colonize the solar system and eventually the galaxy.*
* Assuming past performance is indicative of future performance.
> a human can circumnavigate the planet in less time than it takes light to reach the nearest star
I was surprised by this, but the math checks out (assuming you mean the nearest star other than the sun of course). The circumference of the Earth is about 40000km, at a leisurely pace that's 10000 hours or about 400 days. Proxima Centauri is about four light years away, so the walker has plenty of time for sleep and sight seeing.