> This is not sustainable.
This is a longstanding concern going back to at least the 70s [1]. The argument is essentially twofold:
1. Population growth will continue at or near present rates indefinitely; and
2. We will exceed population N where N is the number of people the planet can support.
For (1) the rate of growth has slowed and is projected to continue slowing over the next century [2] such that many of these estimates have been way off.
For (2), we have a pretty narrow-minded view of how many people this planet can actually support but with sufficient automation hydroponic yield depending on crop can be 10x+ per unit of land area (eg [3]). Automation here is important for both constructing greenhouses and maintaining the crop. This could well increase food capacity to 50-100B people depending on diet, possibly much higher.
What's more of a factor are resources and in particular energy costs. Commercial fusion power with deuterium would solve a whole bunch of problems at once (eg if the cost is below hydrocarbon based fuels then you can extract CO2 from the atmosphere and turn it into fuel, which also largely solves your greenhouse gas problem too). If we ever figure out commercial fusion based on normal hydrogen well that's basically unlimited energy.
Fusion isn't a requirement though. Solar gets ever cheaper. At some point it will become economic to put solar collectors in orbit and I've seen estimates for the yield of this to be about 7x the energy production of the same panel on the ground since there's essentially no night and there's definitely no weather.
Don't get me wrong: people are screwing up the planet and a lot of the larger animals seem doomed but that's more an issue of human vanity (eg wanting rhino horns, ivory, fish bladders and the like) rather than being driven by any actual need.
Deforestation is an issue but more-so in the developing world and more of an issue for ecosystems rather than, say, the atmosphere (IIRC ~80% of the CO2 is produced by the oceans). Overfishing too is a problem but mainly for us because at some point there just won't be any fish to eat.
People like to make alarmist predictions. In some cases I'm sure they truly believe them. This to me is largely a folly of youth as experience teaches you the poor track record of such prognostications. In other cases I suspect people are trying to prompt action from the complacent. The problem with this is that alarmist rhetoric about the environment tends to turn a lot of people off. And a history of bad predictions doesn't help your cause (whatever that cause is).
People do and will continue to make short-term decisions based largely on self-interest. That's just human nature. In a lot of cases its based on need rather than want. It's hard to tell a poor family in a developing country not to chop down the forest when the alternative is not heating their homes (with the wood) and/or not feeding their families (with the farmland created) In others, it's not.
But the notion that this planet can only support 5, 7 or 10 or 15B people is patently false.
[1]: https://en.wikipedia.org/wiki/The_Population_Bomb
[2]: https://en.wikipedia.org/wiki/Projections_of_population_grow...
[3]: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4483736/