Civilization collapses if 50% of people die; unless perhaps if you selectively make those the youngest and oldest of society, the children and elderly that do not work.
But even then we have only delayed the problem, we have no young to fill the worker pipeline, as the workers age and die or retire, so civilization collapses anyway. Back to a state of low production.
But, presumably, after the wars and revolutions, that will sort itself out again. Humans can, if they want, reproduce exponentially, in early America six and eight kids for one breeding couple was not unusual at all. So in a single generation, two adults replace themselves with six adults, 3x as many.
So of course a 50% or even 75% reduction in population, although socially, politically, economically and likely technologically catastrophic, could be back up to 9 billion in a few generations (allowing for much greater childhood and elderly mortality), say 50 years or so.
Since the short-term carbon cycle is roughly the same length, 50 to 100 years, if the survivors did not change their ways in producing energy, we'd only see a 50-100 year delay in the climate change catastrophe: It would just come back; because a century is not enough time to pay down the carbon "debt" we have accumulated.
And face it, they probably would not change their ways. In the collapse of civilization, you burn wood, coal, oil, trash, whatever to generate energy and survive.
And people that rely on the current infrastructure, products, medicines, medical care, law enforcement, delivery services and so on will be dying by the hundreds. Businesses and governments do not scale to half or a quarter their size and just keep operating. The bigger they are, the less they can scale down and still survive. Big companies are intricate machines made of people, you cannot just arbitrarily remove half of the parts and expect it to continue functioning as it was. (If that were possible, the Big company would fire half their people and pocket the profits; right?) Nor can you remove half their customers, and thus half their revenue, and expect them to just chug along. They have fixed costs, mortgages and loans and property taxes, and very few companies have profit margins of over 50%.
All the big companies and manufacturers and food producers and transportation businesses fail. Completely dysfunctional because of their lost critical workers and broke for lack of sales. Practically overnight.
Government and law enforcement and police forces hit the same wall, dysfunctional for lost critical workers and no revenue to pay the rest.
Anybody that relied on products (like medicine, insulin, pills) or services to stay alive is suddenly out of luck. People will form tribes with their remaining neighbors, family, and friends, to protect themselves and find a way to survive.
All of this gets worse if we lose more than 50%. In the Black Plague, at least, most people were much more self-sufficient than we are now, on isolated farms, or even hunter-gatherers, where scaling down is not as devastating. Fewer workers mean less crops but less mouths to feed. There was far less of an infrastructure that civilization relied upon; no power grids or highway systems, electricity or engines or machinery that life depended upon. No factories we bet our lives on. Very lax law enforcement. Primitive medicine, at best.
And that is where the survivors are headed, a setback to our pre-industrial middle ages, with a corresponding population.
But the knowledge of how to industrialize would still exist amongst the survivors. You'd still have engineers and professors. So the rebuild would start, and likely be complete within a century, without having paid much of our carbon debt, so climate change would be recharged and pick up again.