Does the hangover pill work?
To be honest, I didn't pay too much attention to the pills at the time. The couple I took slipped down the throat easily enough and were instantly forgotten. I really didn't think they would make the slightest difference. Apparently, the formula was devised at the Russian Academy of Sciences and was launched commercially last year by a company based in California. I am afraid that this information caused a fierce outbreak of scepticism.
"The couple I took slipped down the throat easily enough ..."
Does this sentence make sense without the throat?
"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."