Imagining a scenario where the clock is running and the end of the game and the team with the ball needs to kick to win: can the head coach ask the referee to apply his timeout at :03 or does he have to manually make the timeout signal?
We seem to always see them make the signal, but is there a scenario where the coach can say “if it gets to a 3 seconds, I want a timeout?”
Or do they just have to do it every time?
IMO teams should have a button to push that will signal a TO. There are plenty of clips of coaches furiously trying to signal a TO and not being able to get the attention of a ref.
Strictly speaking, they always have to signal it. However, you might notice the coaches will grab one of the officials and let him/her know that the coach will want to call a timeout. This is especially true when the coach is deliberately letting the clock run down. The coach may even say that they’re planning to call a timeout with X seconds left, but they still have to actually call it.
So to answer your question directly, the coaches always have to formally call a timeout and make the signal; they can’t tee one up and rely on an official to call it for them.