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Somehow, this sound more interesting than sending Buzz Lightyear to Mars.

OK, a Mars mission would be pretty cool, but this would be some serious science. Can you image any planets in the system of the 3 closest stars? What are their spectra? Can we confirm any inferred orbital periods with direct observation?

You don't think a mission to Mars is "serious science"?
Meta: I like how both you and I were down-voted. Even though we have opposing opinions. Whatever.
It is, you're right. But finding out something about the chemical makeup of an extrasolar planet seems much more interesting to me, especially if we find a "goldilocks zone" planet, regardless of whether or not there is evidence of (oxygen generating) life.
While this would be serious science, I'm not sure why you believe a manned mission to Mars would result in less. It just wouldn't be the same.
Looking back at the Moon program, it was not a science goal--it was political, and a boon for the defense contractors. Now, did some real tech and science benefits result? Yes of course, some of both, but that wasn't the goal really.

Similarly, feet, flags and colonization on Mars is a straight engineering problem. If we only wanted to do science there, we should iterate on the rovers and get more done sooner, for less.

That said, I think we need to do both the OP astronomy and the colonization!

> Similarly, feet, flags and colonization on Mars is a straight engineering problem.

I agree with this and the rest of what you said, but the engineering isn't everything. We'd have to get fully sustainable food supplies on Mars, and as far as I know that's not a solved problem (and, if it is not, it's one that might require knowledge we don't have yet). Feel free to correct me if I'm wrong, this is not my area of expertise.

I think the point was that thing like sustainable food supplies on Mars is an engineering problem, not a basic science problem. We already know how to grow food given water, atmosphere, temperature, and light, no basic science needed. However, since Mars is lacking all those things, the engineering problem is figuring out a way to get them. It's possible to do that with what he have today (e.g. hydroponics with artificial lights), just not easy or cost-effective.
What this misses is that it provides selection pressure for scientific research that can solve these problems in a better way than our current ability. Not all engineering problems end up solved in a way that could be envisioned at the start.
We know how to grow food on Earth. We don't yet know if there is something on Mars that would change things. Until we actually do it, all we have are educated guesses.
What I am missing from the discussion and have not seen addressed on their website is whether this project actually makes sense. I thought that simulations have shown that there are no stable orbits around AlphaCentauri (A&B) [at least in the habitable zone] so what do they think they will find? I mean if we are pretty much sure that it does not exist what they want to find, why do they keep claiming that that is what they will find? There is a reason NASA stopped projects in this direction.
There have already been two planets detected (and possibly two others), and there are at least some simulations that indicate that a planet could be in the inner edge of the habitable zone. [1]

[1]: https://arxiv.org/abs/0811.0673

Even just getting their spectra should be enough to detect life, according Lovelock's theory that planets having life have out-of-equilibrium atmospheric gas profiles.

For example, we should expect that Mars is barren of life, or at least has such tenuous life as to be very hard to detect from Mars' atmosphere.

I do not think so. the spectra from the europa's vapour plumes does not show such signature. But still experts think there might be some bio-chemical reactions going on in the europa's sub-surface ocean.
How long until all the funds are spent and there are no results?
This related NASA idea may interests folks in this thread:

> The New Worlds Mission is a proposed project comprising a large occulter in space designed to block the light of nearby stars in order to observe their orbiting exoplanets. The observations could be taken with an existing space telescope, possibly the James Webb Space Telescope when it launches, or a dedicated visible light optical telescope optimally designed for the task of finding exoplanets. A preliminary research project was funded from 2005[1] through 2008 by NASA Institute for Advanced Concepts (NIAC) and headed by Webster Cash of the University of Colorado at Boulder in conjunction with Ball Aerospace & Technologies Corp., Northrop Grumman, Southwest Research Institute and others. Since 2010 the project has been looking for additional financing from NASA and other sources in the amount of roughly US$3 billion including its own four-meter telescope,[2] or $750 million for one starshade to be used with the James Webb Space Telescope.

https://en.wikipedia.org/wiki/New_Worlds_Mission