As per the plot your village is on the brink of extinction even before the Enclave attacks and kills a bunch of them. The Vault people are killed and kidnapped as well. Your initial population is, optimistically, a few hundred people.
The games aren't super specific with what a GECK even does, but it primarily seems to be about terraforming the earth into a fruitfull state and giving you the basics to start your colony with.
So your few hundred people will be humble farmers like the inhabitants of the original Shady Sands and even massive birthrates aren't giving you a 'metropolis' for generations.
Remember the original Shady Sands in Fallout 1 was build from a GECK, and they couldn't even fight of Raiders.
Vault City was built from a GECK and they're a lot more advanced, but they can't project power and rely on slavery for manual labor.
You can argue, if you like, that Arroyo is just going to be much better than other societies we see at integrating outsiders, and that is how they grow. But a confrontation with the NCR would surely happen within a generation or two of the games ending, and the amount of growth, integration and productions we would need to see for our scrappy villagers to stand up to the expanding superpower of the NCR is incredible to me.