I've seen this argument before, and the problem is, the problem is not just about population.

You can make the case that birth rates are lower in more developed countries, because that's true. But the "more capitalism" argument misses that individuals in developed countries consume way more those in developing countries, and more unsustainably. Globally, the average carbon footprint is close to 4 tons per person. In the US, it's 16 tons. So a family of four, five, or even six in a "poor" country is even with an individual in a "wealthy" country. The problem has not been solved, there's just fewer people consuming as much stuff.

The problem with capitalism, as it relates to ecological overshoot, is this production and consumption. An economy of "stuff," with a wildly unsustainable supply chain. More global consumer capitalism does not fix that.