"Slipped down" is not particularly associated with swallowing -- many things "slip down."
But the passage has already specified that we are talking about pills, and the sentence says "the couple (of pills) I already took slipped down. . "
Under the circumstances, it's hard to imagine any other place except "the throat" that "pills you are taking" could slip down, so I think it would be unremarkable if "the throat" were deleted here.
Possibly pills you were PLANNING to take could slip down the drain in the bathroom sink, or slip down between the cushions on the sofa. But this passage clearly means that the pills were successfully swallowed, not lost, so those are not reasonable ways to construe "slipped down" -- even without "the throat."
- For Teachers