It’s very unlikely that we could cause our own extinction through climate change.
Probably the most likely method for causing our own extinction would be a super bacteria created for the purpose of biological warfare (we’ve already done some pretty nasty stuff in re bio weapons, imagine what a few hundred more years of technological advances might give us!)
Many scientists now believe that the eruption of the Toba supervolcano around 75,000 years ago nearly wiped homo sapiens off the planet, reducing us to around 1,000 breeding pairs. This theory is offered up as an explanation as to why humans are one of the least genetically diverse of the major mammals, every human alive today is related to a small handful of humans (around 1,000-10,000 people) who survived the explosion.
There’s nothing from the Neolithic era on that can really approach what happened when the Toba volcano erupted. Toba erupted around 2800 cubic km of volcanic material, the largest volcanoes to erupt in recorded human history (like Mt. Tambora in 1815) erupted around 100 cubic km, so the Toba eruption is around 28 times the magnitude of Tambora. When Tambora erupted it created a “year without summer”, Mt. Tambora is in Indonesia and as a result of its eruption, there was no summer in 1815 in North America.
Global warming (although estimates vary) suggests an increase in Average Mean Temperature over the last 100 years of around 0.6 C. After Toba erupted it is estimated Global Mean Temperature dropped at least 3 C in a few years, and caused catastrophic climate change across the entire planet. There is a layer of Toba Ash on the Indian subcontinent (thousands of miles away) that is 20 feet thick.
The thing that is bad, for humans and other animals is climate change, a warm climate isn’t bad, it’s the transition that could be killer (we’re still currently in one of earth’s “cool” periods.) If you “put humans in a void” and could ask them to choose between our current global temperature and say, the global temperature in the Eocene epoch, picking the Eocene would probably be wisest.
During the Eocene epoch temperatures at the equator and current tropical areas were much as they are today, however there were no “arctic” areas. At the very least the arctic had temperate forests akin to the modern-day Pacific Northwest, some studies have show that the poles may have actually been tropical.
A fully tropical world (one with livable temperatures both at equator and pole) has obvious benefits over a world where above a certain latitude agriculture becomes unfeasible.
Unfortunately we don’t get to choose from a void. Our current civilizations, cities, and agricultural systems are built around what we’ve known over the past ~1,000 years or so. Rapid climate change would result in great hardship, it’s no easy feat to move coastal cities, and because of the advantages of sea-ports many major cities the world over are built on coasts.
And before the Eocene epoch became a global tropics, there was a mass extinction event as a result of the rapid warming. The millions of years of cooling and global-tropical weather was great for the species that survived the change and the new ones that evolved, but the ones that got caught in the lurch probably didn’t enjoy it so much.