It is literally impossible to make a society sustainable via birth rate alone, which is why I object to the argument vigorously. The only solution is to find a way to make our society and technology sustainable, rather than just hoping that birth control will fix it.
And who is making that argument? This isn't what I'm saying at all.
"The only solution is to find a way to make our society and technology sustainable, rather than just hoping that birth control will fix it."
I'm asking is our current global population level sustainable? It sounds like that's a no. There is at least point-in-time population limits to what our "sustainable" system can handle based on advances in technology and societal restrictions. I use quotes here because we do not have a sustainable system, and the number of people who can be sustainably supported is at least billions lower than we are at today given today's technology.
It has to be a combinations of available sustainable tech/lifestyle and having a population that doesn't exceed the limits of sustainability.
At the current levels, it is not. If we shifted over to renewables then it would be.
I find focusing on the population extremely weird, given that the vast majority of emissions are coming from a relatively small portion of rich nations. We could eliminate the entire global south and still tip this planet into disastrous global warming purely based on the emissions of small, rich countries in the north.
Put more blatantly, I find that a lot of the discussion about overpopulation is an attempt to put the blame of global warming and our sustainability problems on those who emit the least per capita. There is very little merit to discussions about controlling population size while rich nations continue to depend on fossil fuels to maintain their standards of living.
How would that help soil erosion, food production/waste? How would that help with other resource use like metals, timber, fish stocks, etc? There is a lot more to sustainability than just CO2. Even looking just at that, where can we get that much lithium, copper, concrete/asphalt substitute, silicon for solar, etc? There are restrictions to the resources we use and process.
"Put more blatantly, I find that a lot of the discussion about overpopulation is an attempt to put the blame of global warming and our sustainability problems on those who emit the least per capita."
Where was that said? That's not what I'm saying. You still seem to be focused solely on fossil fuels here ("emit"). As nations develop, they will also adopt a more modern and higher consumption lifestyle. This is a global trend that also applies to developed nations continuing to get richer and indulge in having an even better life (eg parents always want more for their kids than they had for themselves, substantial numbers of people want the newest iPhone, look at house size over time, etc).
We are expected to peak well beyond our current level. Estimates are about 10-11 billion - a 25%+ increase.
A major problem with your infant mortality comment and consumption comment is that reducing infant mortality is also associated with modern lifestyles and an increase in consumption. Consumption patterns change as the mortality and fertility rates drop and economic opportunities increase. Our economies are built on consumerism and spending. We see this trend in developing nations today (you have to compare for various demographics, not just the country level).