China has built cities basically from scratch in few years, and big infrastructure very quickly, as a counterpoint.
Big industrial equipment is really heavy btw, so you could fit at most 1 or 2 pieces of big equipment in the 100 ton payload of a Starship rocket. It could easily be a few hundred launches just to get a proper scale foundry going on mars, if you take into account all the required auxiliary equipment like ore scrubbers, cranes and forklifts. Don't forget all the solar panels and/or nuclear plants for power generation too. It would probably be at least tens of billions of support from Earth.
All in all, I would think that 5-10% growth rate is already pretty ambitious, leading to a doubling rate of 7 to 14 years. If you start your colony with about 50 people, it would rise to a thousand people in a little over 50 years.
Minor nitpick: There are only very few plants that are able to extract nitrogen from the air. That's because N2 is a very stable molecule and it takes a lot of energy to use it chemically. Almost all plants are dependent on other nitrogen compounds they extract using their roots.
[That's the reason why the invention of the Haber-Bosch process, which fixes the N2 from the air into NH3 (ammonia), was such a big deal. It is estimated that 50% of the nitrogen atoms in your body once passed through the Haber-Bosch process.]
There's probably plenty of nitrogen in various minerals on Mars. Just not in the atmosphere.
So you would NOT want to colonize a planet first and then terraform it. Instead you would terraform it first. Terraforming Mars for example could involve sending a large number of comets to crash to the planet. If there were existing colonies on the planet, they might be destroyed by the impacts or be inundated by the torrential rains.
Isaac Arthur has some thought provoking videos about this.
The sheer population needed to run a complete supply chain for high-technology (on which Mars would be entirely dependent for basic life-support) is staggering. Having the entire thing built and operated underground would be an engineering and social feat of unprecedented scale in humanity's history.
Iamgining doing it in a generation is simply impossible. Its more likely to happen by quantum fluctuation in a second than by humanity's efforts in a generation with current technology.
Global warming, nuclear winter, asteroid hit, massive solar flare, supervolcano, plate tectonics - anything we care to imagine will leave the Earth much closer to being habitable than Mars.
Just playing along...
But how do you know this? Are there any papers or reports that say that a thousand years is the lower bound on a self sustaining colony on another planet?
It seems to me that this idea is a common wisdom that people often say because in their mind colonization is an out of reach idea so they just pick a random big sounding number like "a thousand years" as a placeholder for 'it is impossible" when there isn't any actual research that says it will take a thousand years.
Think about how economically unviable and hard it is to just create 1 new chip fab on Earth. And think about the fact that on Mars you would need to re-create the entire supply chain that already exists on Earth. Remember that the minerals that you need are probably about as spread-out on Mars as they are on Earth, so you will need a global network of mining stations and processing plants to get the various necessary metals - all this without any access to fossil fuels of any kind, and with all of humanity spending little time outside for fear of radiation.
Perhaps establishing a colony that gets the vast majority of its supplies from Earth is doable in a generation or two (though of questionable value). But creating a self-sustaining colony is entirely out of reach for the forseeable future. You don't need actual research to tell you this any more than you need actual research to tell you that humans can't flap their wings to fly - a little bit of back-of-the-napkin calculations will quickly elucidate it.
Not certain to happen but a fair bit more than your quantum fluctuation.
NB I don't think private capital would ever be enough to get a self sufficient colony started.