There is NEVER a requirement to rush EFATO procedures
Err, in multi ops maybe. In a SEP, you need to get the nose down NOW! Low speed, high alpha, low inertia - you need to be quick with forward stick. But it doesn't take thought, just reaction in a Pavlovian manner.
Plenty didn't, and stalled and spun and died as a result. Including a guy I knew in a PA38 a few years ago. If he'd got the nose down instead of sending a Mayday he'd almost certainly be with us today.
FLY THE AEROPLANE!