Let's further breakdown (Emissions per capita) into (Emissions per GDP) * (GDP per capita). We can see here[1] that emissions per GDP have been decreasing as far back as that data set goes (the past 25 years).
So it looks like the two main factors currently driving emissions higher are increased wealth and more population. No surprise that the US has more emissions per capita than India - the standard of living is much higher.
[1]https://data.worldbank.org/indicator/EN.ATM.CO2E.PP.GD.KD
Edit: linked the wrong chart.
It's not wealth and population, it's the use of fossil fuels (primarily) and, to a lesser extent, ruminants (cows, sheep, etc... both methane emissions from gut bacteria and land-use for grazing). We can have ~10 billion people at US standards of living if we solve the technopolitical problems that keep us on fossil fuels.
It's all three. Our formula was emissions = (GDP/capita)(emissions/GDP)(pop). Reduce any of the three factors and emissions drop.
Your phrasing is off. You can't say "it's not factors 1 and 3, it's factor 2" as that implies factors 1 and 3 don't affect emissions. We could choose to focus our efforts on (emissions/GDP). That doesn't mean the other two factors don't affect it. Reducing human population 10x would have the same effect on emissions as improving our (emissions/GDP) 10x. Spending $10k/year instead of $100k/year will almost certainly reduce your carbon footprint.
France and China are a good example of why the other two factors might be important. Looking at their wealth (GDP/capita (PPP)) and emissions, we find that France is about 4x more efficient in emissions/GDP. But France is one of the top countries in emissions/GDP and China is one of the worst. If we assume the whole world becomes as efficient as France in emissions/GDP and the population increases to 10 billion and the world average GDP/capita increases to match the US (~3.6x increase), we end up with about 1.2x our current emissions.
So you could focus all your efforts on trying to make the whole world much more efficient than France in emissions/GDP, or you could look at the trade offs between all 3 factors and try to reason about how we'd like to balance those factors.
Over the last 200 years the population has went up by a factor of 8. It's hard to nail down exactly what factor emissions/capita has went up since the number used to be so insignificant but you can compare countries right now and see the US has more than 10x emissions/capita than India and higher overall emissions output which is the opposite of what you'd expect if population was the problem.
We could cut population by 99% and say "no more problem" but that's an indirect fix to the main cause: increase in emissions/capita.