The rest of the article answers that question. The followup article answers it more directly, and compares polar to rectangular. https://blog.szczepan.org/blog/monte-carlo/

Short answer: yes it’s uniform in area. In the absence of the specificity you want, area makes the most sense, right? Uniformly sampling independent Cartesian variables yields uniform sampling in area, unlike polar where a uniform sampling of the independent variables gives you a non-uniform sampling of area.

I don’t understand what you mean about it not being an area problem, but I guess at some level this actually is an area problem. I’ll speculate wildly there might be a way to transform the question/setup into a different but equivalent problem that can be directly visualized as solving for area, and perhaps have a more intuitive solution that involves fewer determinants of Jacobians. Maybe, maybe not, I dunno.