Note that I am personally fine with suspending my disbelief on this one (which is unusual, I am picky on this).
That said, yes, asteroid impact causing Venus-like runaway global warming by itself makes no sense, for the simple reason that Earth has been repeatedly impacted in the past and it never happened once. Proof: we are here.
What could have happened with the subsequent meteoric winter is a runaway Snowball Earth. Though it may not have been caused by an impact,
Snowball Earth happened in the past. It may even have frozen the entire surface of the planet, with liquid oceans stuck under the ice sheet (some think that a small band was still unfrozen around the Equator, though). The interesting thing is, as the surface is frozen, white ice makes albedo rise, so more energy from the Sun is reflected, so the planet keeps getting colder until a new equilibrium is found, in reverse of the ocean-evaporating runaway warming.
The problem with Snowball Earth is, it won't kill Earth for good. In fact, in the Cataclysm text, one brute-force technique is referenced: covering the ice with black soot to lower its albedo, making it heat up and melt.
Previous Snowballs probably melted due to accumulation of greenhouse gases released by volcanism (idea for softer SF: a volcanic ice planet!), so here is another proven brute-force solution to the problem.
Now, would it be possible to cause a Venus-type runaway warming with asteroid impacts? Here is what I would try: take a comet, set it on impact course and break it in little bits (which, incidentally, are harder to deflect that one big chunk).
The comet being made mostly of water, it tends to explode in the atmosphere, which may be what happened with the
Tunguska Event. If you are trying to flatten a country, it is also better to have multiple high-altitude explosions, the way city-killer ICBMs will use multiple independent warheads set to explode well above the ground in airburst detonations, instead of one big warhead exploding at impact. This way, while the impact point takes less damage, the surface that takes enough damage to raze cities (and, by extension, countries) is greater, which limits overkill. A fragmented comet is simply this many times bigger, with the added bonus of no fallout! I am sure this will make things better on the long run.
Then what happens when you suddenly add a pile of energy to a chunk of the atmosphere, in the form of heating and anarchic movement? In addition, you just added a pile of water - not much compared to the entire Earth hydrosphere, but maybe enough to further unbalance things.
Well, climate scientists could build models and make predictions with varied probabilities, but in the end, no one can tell for sure. And don't try that on Mirror Universe Earth to see what happens, as there is no guarantee that it would have the same effects each time. So you can go and say that it caused runaway global warming and no one can say that it couldn't happen. Particularly as, contrary to good old asteroid impacts and probably the occasional comet, such rain of cometary fragments probably never happened in the past: it is not something that would naturally happen.
I still have a softer spot for Snowball, with its cities, highways, farms buried under the ice. Maybe a few survivors, or archaeologists digging under the ice. But Boiled Earth is more definitive, and I am not sure "Children of a Dormant Earth" is a better title.
