Re birth rate.
This is predicated on two things, the first is that we do nothing to reduce per capita emissions world wide, in which case obviously more people = more emissions. However every single country with any credence has committed to carbon reductions, most, including China and India, have also made net zero commitments. https://eciu.net/netzerotracker This completely defeats this element of the argument for population reduction.
The second is a really dodgy report commissioned by the otherwise very well respected Climate Disclosure Project (CDP). The report allocated emissions from individuals descendants, for something 20 generations, to the individuals alive today. This goes against every carbon accounting procedure (which CDP were instrumental in drawing up). Again it assumes "business as normal" i.e. no per capita emissions reductions. Which undermines all the assumptions it's based on. I've tried to find the report just now but it seems to have been taken down, although no doubt the alarmist headlines are still out there in the media.
A similar report concluded that twenty companies (or at least very few) were responsible for 90% of global emissions. However this allocated all the emissions from e.g. you running your car to BP or Shell. Again this is ridiculous as it absolves individuals, companies and other organisations of any responsibility. Makes great headlines though!
Far too many poor reports are just looking to generate headlines, what they seem to have in common is miss-allocation of emission scopes, i.e. scope 3 as scope 1, and an assumption of "business as normal" with no per capita reductions in emissions.
Generally the reporting on climate change is really really poor. It's a really complex subject with many many feedback loops and interlinked mechanisms but basically:
We need to stop burning fossil fuels!
Link to scopes here
@Hiker50 great positive podcasts Radio 4 39 Ways to Save the Planet (I'm in absolute awe of the mine shaft battery!)