> Look how small and insignificant we are ..... It seems almost nihilistic.
It isn't nihilistic. It is a simple fact.
As an aside, if you have not yet listened to the Hitch Hikers Guide to the Galaxy radio plays (and you should!) you will know this is a major plot device.
It is a simple fact that the Earth and solar system are small relative to the universe, because everyone agrees on how to measure dimensions and volumes. It is not as simple a fact that something more large is more significant -- whatever that means (if we knew what it meant, this claim would be settled).
Physical size is such a universal metaphor for relevance/significance (eg, "it's a big deal", "it's a small matter") that they're implicitly treated as synonymous.
There is a well defined meaning of "insignificant" in this context: none of our activities have any material impact on what we see beyond the confines on Earth.
So the precise claim is that to describe any interplanetary scale (or up) process you can completely neglect humanity in its entirely, because its contribution to the process is negligible.
You are right that there's an unspoken step from "too small to impact the sun in any important way" to "insignificant to the universe", but it is not actually a very large step: being unable to affect one's surroundings is a classic sign of powerlessness.
We're insignificant now. If we develop interstellar travel, we could start to become significant if we the visit/colonize the rest of the galaxy. Of course, even if we do fill up the galaxy, our galaxy is only a small sliver of the observable universe, so we'd have to move on to other galaxies. Not as crazy as it might sound, if given we would have the tech for interstellar travel, as intergalactic distances are only ~3 magnitudes off the span of a galaxy. e.g. Milky Way spans ~30k light years, Andromeda is ~3m light years away, 100 times. The leap from short hops of a couple light years to galaxy-wide distances is greater (Proxima Centauri is ~3 ly, vs. 30k light years = 5 orders of magnitude). Which is comparable to the jump from interplanetary to interstellar distances distances, which is about 4 orders of magnitude.
So if we can reach the edges of the Milky Way, other galaxies should be within reach. But doing that before the Heat Death/Big Rip/etc. might be a challenge.
It isn't nihilistic. It is a simple fact.
As an aside, if you have not yet listened to the Hitch Hikers Guide to the Galaxy radio plays (and you should!) you will know this is a major plot device.