I think half the problem with rapid (unthinking?) responses to non-normal situations comes from the format of some simulator checking/training, certainly in the past. The emphasis was on getting recall/memory items done as fast as possible and as it was the same V1 cut/fire every time, it was easy to get into the habit.
Having been exposed to much of the above negative training, I make a conscious effort to a) fly the aeroplane above all else, b) spend as much time as necessary diagnosing what's happening and c) only take action when we are in agreement as a crew about what needs to be done and there is sufficient capacity available to effectively monitor it.
There are very few things that need an immediate response: high speed RTOs and hard GPWSs come to mind - for pretty much everything else there is time for some thought and interaction. When you get an engine fire warning, does it mean that the wing (or wherever the engine is mounted) is a mass of flames? No, it signifies that the temperature inside the nacelle has gone over a pre-defined limit, not an amazingly hot one either. It could be a false warning, although with multiple loop systems, that's unlikely.
My company has 400' as the minimum height to initiate any drills and I emphasise in briefings that it is just that, a minimum, not a target. If it takes us until we're clean (or later) to diagnose the issue(s) and start actions, then that's not a problem. Getting the aeroplane away from the ground, in trim with as much automation in use as possible is the primary goal. After that, making sure that the projected flight path is the one we want, especially when OEI. Only when the previous two conditions have been satisfied is it sensible to start recall items.