What is the most common practice of generating (X,Y) from an arbitrary CDF or PDF?
I am no expert in the field but I can imagine 3 different cases :

Variables are independent in which case you can use your formula on each
You have access to some chain rules decomposition that allows you to easily generate each variable one at a time using your formula
If none of the above : https://appsilon.com/how-to-sample-from-multidimensional-distributions-using-gibbs-sampling/