I think we have to distinguish between V1 cuts, where large and timely rudder deflections may be required to adequately contain the yaw and failures at higher speeds.
If you had the critical engine run down during a close-in turn after takeoff, flying manually, how would you a) recognise the problem and b) deal with the short-term effects? I put forward that unless it was a sim training detail where you were expecting something like this to happen, the first clue would be an increasing tendency to hold off bank, followed by reducing airspeed as the pitch attitude would be too high for the remaining thrust. The initial response of most pilots to a roll deviation is to apply opposite roll control, as 99.999% of the time it is caused by turbulence, wake or otherwise. The first response to a trend of airspeed loss is to lower the nose to a more appropriate attitude.
After a short period of
the amount and duration of the aileron input required would lead most people to feed in rudder to assist, especially if the roll input was a significant proportion of the total available. At that point, there would be enough clues, including engine instruments now showing a problem, that would trigger recognition of engine failure and the appropriate responses and drills.
Point is, it doesn't require violent rudder inputs a split second after the event to stop the aeroplane flipping onto its back. Applying a measured amount of corrective yaw in response to what you have recognised as an unwelcome trend I'd say was a better solution...
Coming back to the central theme of this discussion, I agree with many of the posters that there is no need for significant rudder inputs in jet airliners, outside of low-speed engine failures and crosswinds. The only possible exception that I can think of is if you had reached the limits of control and were heading outside of them, then you could (gently) try the rudder *as a last resort*. At any more than moderate speeds, the primary and secondary effects of rudder input are so powerful that you stand to lose much more than you could possibly gain.