I wasn´t aware that Stoppard wrote KOTCS.
Well, he didn't really write KINGDOM OF THE CRYSTAL SKULL as we know it. Stoppard did some work on INDY IV, but I believe he did it
before Frank Darabont came on board, somewhere around 2002, when INDY IV was still in a relatively early stage, and so it's hard to say what Stoppard brought to the table on this particularly project. It's my guess that the script Stoppard turned in doesn't bear a strong resemblance to the final product.
And even after Darabont turned in his CITY OF THE GODS scripts, there came a steady stream of rewrites from various sources, and the project continued to evolve. Lucas himself took a stab at a script (I think it was called PHANTOM CITY OF THE GODS at that point), then there was Nathanson (ATOMIC ANTS), then came Koepp, who whipped it into its finished shape.