> a closed-circuit life support system capable of keeping a human occupant alive indefinitely, for many years at a stretch, with zero failures and losses, and capable where necessary of providing medical intervention. Let's throw in a willing astronaut (the fool!) and stick them inside this assembly. It's going to be pretty boring in there, but I think we can conceive of our minimal manned interstellar mission as being about the size and mass of a Mercury capsule
Perhaps the most effective means of interstellar travel is bases within asteroids, not spaceships. Future humans could build habitats inside asteroids, then with a few nudges upset the entire gravitational equilibrium of the asteroid belt, slinging some asteroids out of the solar system. We're probably able to compute the most optimum way of doing this within the next century, and humans are more likely to survive interstellar trips buried inside asteroids than in a spaceship's husk.
> you're not going to get any news back from the other end in less than decades.
> transporting our Mercury-capsule sized expedition to Proxima Centauri in less than a lifetime.
Why not one-way trips? The new civilizations might consider themselves new "countries", unworthy of interference by Earth in their internal affairs.
I should add we have another billion or so years of life left in our Sun, before we really need to leave our solar system: that gives us enough time to learn how to do one-way interstellar trips. I concede that in the meantime, Earth's population is likely to suddenly drop a few times. The projected 9 billion inhabitants in year 2050 seems unstable, though new food/fuel technology may come before then.
As for mining the asteroids, perhaps it'll be done by Earth-based avatars. Certainly the Moon can be mined that way, with its 1 second response time, but Mars would be more of a challenge, requiring machine-human cooperation in the processing.