Agree with you on the general idea that dimension reduction is a very important tool. But from my experience performing a PCA first and then trying to make sens

Agree with you on the general idea that dimension reduction is a very important tool. But from my experience performing a PCA first and then trying to make sense of the coefficients is a level 100 reverse engineering task. The coefficients will most likely be numerous and noisy, making sense of them will be impossible. You need to have an hypothesis first of what the principal components might be, and only then compare your idea with the PCA coefficients to see if that works.