I think any deeper meaning we’re going to find is going to be hard to come up with and I am going to chalk it up to “the writers didn’t really consider that.”
I’ll easily concede that the real answer here is likely: “The writers were looking to make an interesting and compelling bittersweet story about sacrifice, and in that they succeeded. They did enough world building to establish the premise without losing track that the point was to tell the story with the 35ish minutes of screen time, and without resorting to too much exposition.”
Now that we’ve established that, lets explore some plausible reasons to explain it ourselves.
If it is just a rural village, why take one of the bumpkins to represent your entire civilization?
First, we don’t know they only launched one rocket. Perhaps they launched hundreds each in different directions and the probe that Picard encountered was the only one remaining.
Second, perhaps there was only the one (or few) rockets because the world was subdivided into different countries and others valued wealth and power and consumed themselves in hedonism while our protagonists were something like Quakers that had a distinct view on life and different values.
If we were to pick the memories of one person in the entire world to represent our whole planet, do you think we’d go to Podunk, Arkansas?
Well, we kind of did in real life.
The golden record we sent out in the Voyager probes contained only two works from the, then, modern 20th century with the rest being hundreds of years old or tribal works with roots older yet. Here’s the list.
Many works of fiction are “slice of life” which have a story whose main thrust is just to give the reader an immersive experience in the culture at that time in history. Twain’s Tom Sawyer books or Alcott’s Little Women are good examples. The story in the episode is that the residents felt it was more important to keep alive who the people of of Kataan were, instead of what they’d accomplished. That seems plausible in storytelling for me too because any race advanced enough to find the probe wouldn’t be impressed with the technology.