As always, fly the aircraft and don't just do stuff without thinking first.
Absolutely. There are not that many of what some call "time critical scenarios" and only a handful of those need *immediate* pilot input, such as RTOs and hard GPWS warnings.
On my first jet (737-200), the company SOP was TOGA thrust on the live engine after engine failure. We then started operating FMC-equipped variants (300/400, etc.) and had done so for quite a while when someone new to the fleet pointed out that this was not such a good idea as at high derates and low weights, as you could end up very close to, even possibly at or below Vmca, due to the way we calculated the speeds. Cue overnight new SOPs.
Engine "fire" warnings: all they tell you is that somewhere in the nacelle, the temperature is high enough to have triggered the detection system (or there's a fault in the wiring). It doesn't often mean that flames are gushing from every orifice and the engine and/or wing are about to burn off, although this is a possibility. Unthinking and/or panicky reactions to warnings can often put the flight in substantially greater jeopardy than a more considered approach...