De facto, everything you write is correct assuming that the missed approach is performed at or above the DA.
The problem is below the DA, then all bets are off and there are no longer any requirement that have to be meet according to regulations. If e.g. a runway incursion occurs and you have to make a go around or balked landing, you most likely wont have any problems following the standard missed approach with 2 engines. But if at the same time a flock of birds lift of an get ingested in one engine at the same time, your starting point and altitude for the missed approach is very different than what is assumed for the calculations. If there is already a contingency go around procedure it wont help. As this assumes that you engine failed before the approach and that you go around when reaching minimums as well.
In short, with engine failure and go around below minimums - the engine out procedure for the departure will most likely be the safer choice if operating around the limits.
But how likely is this scenario is another question. It would have to be special circumstances before I would waste my time including this in my briefing.