Thought I'd share one of my favorite short stories (written by Ted Chiang, most recently of Arrival fame) that bears on this: https://www.nature.com/articles/436150a
Wait. This device can be fooled easily: At any point in time, I can look at the LED. If it's off, I press the button exactly 1 second later. If it's on, I do nothing.
What am I missing?
You're simply missing the fact that by definition there won't be any case where you see the LED being turned off 1 second before you push the button. If you see the LED being turned off, you won't be pushing it the second later.