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?
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.
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.