I am not leaving anything out, I am using the same table that the BI article used, the one with the $102,000 counterfactual representing "Full-Year, Full-Time, Prime-Aged Workers". The one you mentioned yourself.
If I were to instead use Table 2b, the median counterfactual would only be $57,000 and I would be making the same argument with different numbers.
> In the paper I didn't find figures that I could use to reconstruct how much money would have to end up in someone else's hands to achieve the counterfactual.
All the figures are there. Just take the delta for every percentile and add it all up. If the sum is negative, you can't achieve the counterfactual by income redistribution. (Spoiler: The sum is negative)
Why the huge disparity? It's because the GDP and income are only loosely related. The US has collectively been spending more than it earned for decades, and all that spending goes into the GDP.
> I think BI properly quoted the article, but there would be a lot of work to do to contextualize the whole thing.
They may have "properly quoted" the article, but they came to a wrong conclusion that feeds into the popular "the rich are taking all our money" narrative.