I’ll probably trigger some people by saying this, so I made a throwaway. It’s because she’s the best woman of color they could find who was remotely qualified for the job. The Biden admin has been preferentially appointing women of color across the board. Look at the demographics of their judge appointees, for instance[0]. I’m not saying this to be inflammatory. There’s crystal clear of evidence of preferential treatment.
For many “in the know” in this area, Khan has repeatedly made a fool out of herself, despite those who wish to push a glowing narrative. As many have already pointed out, she’s simply not that effective, and at some point you have to admit that even if you agree with what she’s trying to do (as many of us here do). One of the reasons here, which you mentioned, is the remarkably low staff morale at the FTC which she has caused[1].
Prejudice like this has been happening across the board under Biden, and the vast majority of the picks are, to say the least, underwhelming (Khan isn't really that bad relatively speaking.) Bottom line is: if you or anyone reading think this is crap, you need to get yourself (and others) politically active.
[0] https://www.pewresearch.org/short-reads/2022/08/09/biden-has...
[1] see, for example: https://www.washingtonpost.com/technology/2022/07/13/ftc-lin...