Indeed. I look at the effort required to get a teeny tiny solar powered helicopter to Mars, and while the achievement itself is noteworthy the difficulty in doing so is illuminating. Dreams of human colonization of Mars are largely coping mechanisms to help with the fears that arise when considering the very real apocalypse humanity faces.

> building infrastructure on Earth takes forever

China has built cities basically from scratch in few years, and big infrastructure very quickly, as a counterpoint.

Yes, but only after they spent decades on building out their construction industry with lots of imported equipment from elsewhere. Mars will not be able to construct big roads and buildings until they bootstrap a concrete industry and they can't do that without a steel industry, and they can't do that without mining sites, which need vehicles and mining equipment, etc etc etc. All these things will need either large amounts of people (and thus food) or large amounts of automation (and thus microchips etc) and Mars will not be able to bootstrap either without billions and billions of investment from Earth.

What's the minimum amount of mass/volume of equipment needed to bootstrap, and what is the doubling rate of this infrastructure?

Yes yes, we all know that exponential growth is pretty cool. But take into account that you can't explode outwards in the Martian environment like some algae growing in a food rich environment, every new person and every piece of infrastructure will need (a lot of) maintenance and that slows down doubling rate by a lot. You also can't use a lot of Earth-based shortcuts either, since there is no oxygen for fossil fuels and you won't be able to grow food in open atmosphere.

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.