That's the received wisdom. Is there good evidence that it's true? Or is it just a widely held belief?
All natural systems are in decline; collapsing.
Water tables are dropping. Topsoil depth is decreasing. The sea is overfished, polluted and awash in trash, its pH is decreasing and temperature increasing. Marine nurseries like mangrove forests are now luxury hotels. Petroleum reserves are becoming increasingly hazardous to tap. Atmospheric CH4 and CO2 are increasing. Arctic snowcover is melting, along with regional albedo. Permafrost is melting and the tundra is outgassing even more Co2 and Ch4. Greenland is melting, sea level rising. The African Sahel is desertifying fast. The Amazon, Asian and African rainforests are rapidly diminishing. Species worldwide are becoming extinct, as human population, resource usage and carbon footprint increases. Invasive species are upsetting established ecosystems.
Just off the top of my head...
Almost all of those things boil down to "too much coal and oil have been and are being burned".
Which is true, but not really a function of absolute population. We don't need to burn fossil fuels - and if world population were dramatically lower, it would just delay the inevitable, without a change in technology from fossil to nuclear energy.
If there were only 800 million people instead of eight billion, carbon dioxide levels in the atmosphere would still reach about the same concentration once all the accessible coal and oil have been extracted and burned. It would just take a few more decades.
The planet doesn't have a carrying capacity in any real sense - a huge number of humans can live well without harming the environment too much; Or a tiny number could fuck it up beyond repair.
The question on the table is 'how many people can live comfortably and sustainably on Earth?'. That the number that we currently have are not choosing to do so is a different problem altogether. It's easy enough to see that something needs to change - but it's FAR from easy to see that the absolute population of humans, current or projected, is one of those things.
If people burn too much coal, that doesn't imply that there are too many people, unless it's unavoidable that people must burn coal.
Whaling was unsustainable in the 19th century. World population is now massively greater than it was then, and yet the pressure on whales has gone, and their numbers are rapidly recovering. So while there was clear unsustainability, it didn't translate into a clear example of "overpopulation".
So, again: Where is the evidence that we are above the planet's carrying capacity? What is that capacity, and why? And if we don't know what it is, how do we know we're above it?
Any problem that could be solved without population reductions isn't a carrying capacity problem. Of course, that doesn't mean it's not a problem - but it does mean that bitching about overpopulation is unhelpful at best, and counterproductive at worst.