This is a great question! I think it is correct to say "the emissions impact of consumption is roughly proportional to the amount you pay for it", but if you mean "the amount you pay is roughly equal to the environmental impact measured in monetary terms" then the question is a lot more complicated, but I don't think there's a good reason to think that.

An argument for the first claim is:

1) The supply chains for almost all goods and services involve energy produced from fossil fuels

2) Energy is a variable cost, so is roughly proportional to the amount you pay for a good or service

3) Therefore, the emissions impact per dollar spent will generally fall within an order of magnitude across most goods and services

This argument also works (perhaps to a lesser degree) to other intermediate products such as cement, steel and aluminium, which appear in many supply chains and have very high non-energy emissions.

The exceptions to the first claim tend to be products that depend directly on processes that have high non-energy emissions e.g. flying (fuel burning), food (land use from agriculture). This disparity will become even more stark as we decarbonise the power sector (which is easy compared to industrial emissions).

One way to be concrete about this is to estimate the "Scope 3 multipliers" associated with the various products in the economy, which give the emissions impact per dollar/pound/euro through the whole supply chain. One approach to this is to use input-output tables augmented with direct emissions-intensities for each sector (this is what I do a lot in my job), but there are other product-specific things you may want to account for (this seems like a good reference)

I can't get on to the newint site for some reason so I don't know if they say anything more in support of excluding China, but it doesn't seem reasonable to ignore the progress made in a country that contains ~20% of the world's population. But even if we do that, the progress made at the lower end of the income distribution is still significant, and ignoring it by adopting a higher poverty threshold doesn't make sense to me (bear in mind that there would always be a higher threshold at which progress in poverty would look much weaker, and people suggest different ones all the time).

Of course, the choices about what countries/poverty lines to look at depend on what you are interested in. For instance, if you want to know the impact of global capitalism on poverty, you would not want to exclude China if you think that its rise was largely due to liberalization and trade with developed capitalist economies (I am agnostic about this). My sense of Rosling's main point is that people were/are massively underestimating progress, even at lower poverty thresholds where it should be easier. This is true even if progress at higher thresholds has been less good. Whether Pinker's book is misleading on recent changes in global poverty may depend a lot on how much China's progress can be attributed to 'enlightenment values'.

Also leaving this as a good summary of the debate https://www.cgdev.org/blog/12-things-we-can-agree-about-global-poverty