Yes, an organized extermination of 99% of humanity could be the start of a viable solution. The obstacles could be overcome.
Speaking in super-villain mode, 1% of roughly 7.5B is 75M. That is enough to maintain a modern society if you collect them all together and protect them from the extermination. It is larger than the population of England, and 90% the population of Germany. It is 3.5 times the total population of Norway, Denmark and Sweden put together.
Really, a million people is more than enough to keep a modern city running, so dozens of cities could be established, around the world, to be close to the resources that cannot be moved (climate, arable land, fisheries, mining operations, etc), and could be kept running. We could maintain the infrastructure of those cities and grow the food, produce the energy, run the factories, etc.
Especially with all the world's resources, parts and stockpiles untouched and available for the taking. I don't have to depend on scavenging; the actual factories making things like microchips, tablets, farming equipment, etc can be used, we can keep plenty of people that know how to run them, design them, and program them.
The biggest caveat would be population control, but that can be handled too. I can demand genetic screening and abortion for abnormalities, also for parentage. Have all men and women register DNA, and continue to do so at birth for future generations. Parentage becomes a certainty. For both men and women, becoming the parent of a child costs them each a point, and each person has two reproductive points, after which they are surgically sterilized. I'd let them freeze sperm and eggs before sterilization, so if their child dies before adulthood the parents each receive a reproductive point that can be accomplished by in-vitro fertilization, with the same partner or another (e.g. they don't have to stay together).
It makes no sense for me to keep the wealthy, unless I just need them to finance the operation. I have been around dozens of very wealthy people (My skills are particularly useful to them) and most of them are not highly skilled, not great managers or businessmen, and not that intellectually gifted. The source of most of that wealth is a lucky strike (as in striking gold), with some kind of entertainment, an invention that caught on, being born to a wealthy patriarch, having the looks, charisma, voice, or athletic ability to become a star (which does require some skills but not a lot of intellectual firepower), or just being in the right place at the right time or having personal friends that did, so they got in on the dirt floor of the Microsofts, Intels, Apples, Googles and PayPals of their day: One genius with 500 lucky multi-millionaires in their wake, because they got stock for a penny that became worth 10,000 times as much, with no particular genius on their part. (e.g. Bill Gates first receptionist became a multi-millionaire on MSFT stock).
What I want to preserve is talent, expertise, intelligence and problem solving ability, in hundreds of disciplines, and I want to preserve the ability to teach those things to future generations. So if that turns out to be a rich person, fine, if not, I'll take the John Deere engineer that knows how to keep farm machinery running, and everybody else it takes to continue the modern methods of putting bread on the table.
Most of the world would eventually crumble, which is what we want. Return it to forest.
Most of the people would rot and die. But they were all going to die anyway in the next 100 years, so whatever CO2 and pollutants they cause are just a pulse, exactly what they would have produced by eventually dying anyway, and less than they would produce if left alive: Their day-to-day CO2 production (their carbon footprint) comes to an end, thus saving an average of about 40 years of life. Within a few decades, as the parts of the world we don't use return to vegetation and forest, the bodies of the dead are naturally recycled into what we actually want: long lived trees and plants that absorb CO2 and produce oxygen.
For any particular places that we want to use, the dead can be removed and buried at sea as fish food, eventually helping to recover fisheries and sea life.
The oceans and atmosphere would recover, with 99% less pollution, and 99% more forest to absorb whatever pollution we produce. We don't have to give up fossil fuels, our problem is not exactly pollution, but too much pollution so we exceed the absorption capacity of the Earth. Analogous to salt: Eating a pound of salty potato chips won't kill me, eating a pound of salt could produce hypernatremia, salt poisoning that can give me seizures and kill me today.
Yes, exterminating 99% of people can work. There is plenty of land near the northern and southern poles that will still be cold enough for us to ride out the worst of global warming if it ramps up severely, for a few centuries. (Of course enough heat kills everybody and everything, we cannot recover from anything.)
But I [the super-villain] can't just leave the survivors where they are, that would return them to medieval subsistence living almost immediately, or even hunting and gathering. I need a program (and hundreds of people, but I can hire the few dozen leaders I need to recruit and run them) to identify and immunize those I wish to keep (perhaps without their knowledge). Then I need a plan to either have them already be in the right place at the right time, or a plan to protect them and get them there (to one of my 75 sanctuary sites) once the pandemic begins.
I believe both modern civilization and humanity could survive indefinitely with a population of 75 million people. Or with mandatory birth control, whatever we compute as a comfortable carrying capacity of Earth for humans.
I would likely spread out the space occupied by about double, and make living spaces much larger. Crowded and towering cities are not at all a necessity with modern transportation (electric rail and cars), there are only a very few practical reason for a building to exceed three stories if horizontal space is not an issue; it is far easier to build, maintain, and demolish such structures. (Some industrial or scientific equipment is necessarily very tall, housing it is one reason to go above three stories).