Further, how many governments? How many changes of government? How many opposition leaders who sweep into power promising to cut the wasteful space exploration project who then have to be told the Awful Truth? How many bitter former government leaders who thought they’d be able to determine who got a seat on the Ark but now look like they’ll be out of power?
The number of people who might be saved by a workable “survival” plan will be no more than a few thousand; if such a workable plan were possible. That leaves many millions of people who will not survive, ark or no ark.
As the end draws closer, one may expect the social order to disintegrate. There will be chaos, sickness, and famine. Living through that will only let you be killed by the destruction of the planet itself. Many, many millions of people will have very bad deaths. My plan doesn’t save their lives (neither will yours I suspect) but my plan does allow them to choose the time of their passing and to do so quietly, painlessly, and with dignity. Since nobody can realistically be saved, the only humane thing to do is to alleviate as much of the misery inherent in the situation as possible.
I’m not sure you appreciate just how determined the human race can be. Assume all money and effort now expended on defense used in research on ways to nudge the object, including exploring possibilities of anti-matter, nanotech, moon-based mass drivers and God knows what. Several orders of magnitude beyond the Manhattan Project and the Apollo Program combined, of course, with I grant a 99.99% chance of failure, and global tidal-wave catastrophe even in the event of a successful “miss”.
I’d kinda like to see rockets sent not to the object directly but to smaller asteroids that can be nudged with plausible tech to collide with the object. Find a billion-ton asteroid that is already kind-of headed in the right direction, nudge it with a series of nukes, and make it slam into Moonie McEarthsmash with the equivalent of a 50-gigaton nuke. Rinse, repeat and dig some deep-underground survival bunkers capable of holding 100,000 people for 200 years.
In any case, it’s easy to scale up the impending disaster to turn a 99.99% chance of failure into a 100% chance (it’s not only moon-sized, it’s made of Adamantium, and it hits next week!) at which point I wouldn’t bother telling the public but would just live high on credit for as long as I could.
Nitpick: it’s made of Vibranium. Therefore, it’ll just absorb the energy from the asteroids you cause to collide with it.
One more thing. In the other thread the existance of an Earthlike planet orbiting Sirius is postulated, and the purpose of the space ark is to reach this planet. Except this is a silly goal. The space ark is going to have to travel 8 light years. Barring breakthoughs in our understanding of the laws of physics, it’s going to take the space ark generations to reach Sirius. Except any space ark that can sustain itself for generations while it travels to Sirius could do the same thing except here in our solar system.
In other words, the hard part of designing the space ark isn’t the propulsion system, it’s the space habitat itself. And if we can build a self sufficient space habitat, why waste all that effort strapping antimatter rockets on to it? We could build 10 habitats designed to stay in our own solar system for the same effort it would take to construct 1 habitat designed to reach Sirius or Alpha Centauri. We’d be much better off attempting to colonize Mars and/or the Asteroid Belt rather than spending all our energy strapping antimatter rockets onto the habitat, then strapping antimatter rockets to the antimatter rockets to accelerate the antimatter rockets, then strapping antimatter rockets to the antimatter rockets.
The way to play the public relations is to announce that there’s a planet-killer headed for earth, it’s probably going to miss, but just in case it doesn’t we’re gonna ramp up our space program a teensy bit, just a simple thousand-fold increase. Nothing to worry about really. Half the increase goes to Mars colonization, half goes to planet-killer interception. Because really there’s no way we could tell for absolute sure that the planet-killer is going to hit the earth that far out.
Either way, Sean Connery will save us.
You could also introduce a lotto for space on any colonies built. People love lottos. It gives them a sense of hope you’re not just going to pick your friends.
Other problems include that Sirius is a binary star system, and therefore probably unlikely to have any planets with anything like a stable environment.
However, I think the OP of this thread is pretty clear that it’s a hypothetical: “What if we’re all screwed but could get a small number of people to another world?”. So issues of the practicality of interstellar travel, and the possibility of eliminating the impending threat, are irrelevant to this thread.
That’s the core of the hypothetical. You can’t just declare it irrelevant because it appears practically insurmountable.
You must be fun to watch movies with. “B-b-but how could being bitten by a radioactive spider give one super powers?!?” 
Seriously though, your objection is fine for the other thread. But here, the OP is simply, “what if there’s impending doom, but a percentage of us could escape to another planet”.
A perfectly legitimate hypothetical.
If it helps, imagine this scenario is in the year 10,000.
This is why humanity is doomed in the OP’s hypothetical scenario. Even if the construction of an ark is technologically feasible, it won’t actually be completed. All it takes is for one guy to say “oh no you don’t either” and fly a jet into the construction site, ruining months or years of work. And there will be billions of those people.
No, I don’t think. In fact, I believe that you are deluding yourself with a cherished fantasy. Even your vaunted market will cease to function if money has no value and commerce has a termination date, as it would in this scenario. And with no market, there will be no sales proposals, no budgets, no startups. Humanity will eat itself.