Can population control solve the climate crisis?

Yes and no.

According to the IPCC, if we continue to emit as we do now, we have 10 years worth of carbon budget remaining before we reach a dangerous 1.5$^\circ$C of warming.  We can avoid exceeding this budget by  halving our emissions between now and 2030, and then bringing them to zero by 2040.

Let's imagine an incredibly successful population control strategy, based on family planning, that results in zero births this decade.  By how much would the world's population fall?  According to Wikipedia the average life expectancy globally is about 72, so let's assume that anyone over the age of 62 would shuffle off this mortal coil by 2030.  According to the UN there were 962 million people over the age of 60 in 2017, so there's at most that many over the age of 62 now.  Oh dear that means that the population would only fall by $100\times\frac{0.962}{7.8}$ or 12%, but we'd need it to fall by 50% if it was our main strategy in reducing emissions.

So, population control would work, but not if we relied solely on contraception.  It would require - and let me make this 100% clear, I am NOT proposing this - a cull of the population.  But where to start?  It's fairly obvious that the absence of a person that produces a lot of emissions is more beneficial (to those remaining) than the absence of a person that produces very few emissions.  So let's look at just how much disparity there is.

According to Oxfam between 2010 and 2015 52% of the emissions came from the wealthiest 10% (people like me, I guess).  What does this look like?

On the left you have 100 Go stones representing the human population with the wealthiest 10% in represented by black stones (which have the advantage in a game of Go) and the poorest 90% represented by white stones.  On the right you have their respective carbon emissions.  So, getting rid of the richest 10% of the population by 2030 would actually have us on track to deal with the issue.  Let's get rid of 9 of the black stones and see what happens:

That's right: 1% of the population generates 15% of the emissions.  So if we eat the richest 1% we'll bag 3 years worth emissions reductions! Let's get rid of a few of the white stones now:

It turns out that the poorest 50% of the population is responsible for a mere half of the emissions generated by the richest 1%.

I've heard a lot of people - all from the richest 10% - claim that the big issue that no one wants to talk about is population.  But if they argue that population control is the best way to deal with the climate crisis, they're advocating for their own removal!

Personally, I prefer the idea of wasting less, reigning in our excesses, and switching to renewables.