Note that most of the initial tests required a nasopharyngeal swab which has to go back into your nose several inches. I'm not surprised that the FDA doubts that people would be able to do that reliably on their own given that even trained people seem to get it wrong a lot (1).
The LabCorp test uses a swab that doesn't go anywhere near as deep (2). For a test kit to be widely distributed, you need to have it be drop dead simple to run perfectly.
Given all this, I'm not sure we should jump to corruption as the reason this is the first at-home test.