Regarding (2), there is no way that I know of to predict which data sets will give different results for the two methods. If there were, then it would save a great deal of computation since one could compute the quick old method in those majority of cases where the methods are practically identical. In the sense that I cannot predict which data sets will lead to different results, the old method does fail randomly in a practical sense, though not in the usual sense since the behaviour of both statistical procedures is completely enumerable.

Regarding (1), I am not clear in my own mind what constitutes an incorrect inference for a particular data set. So, I am taking the view that if my P-value is smaller than the standard P-value and is also sensitive to departures from the null (as revealed by power calculations) then my method is “correct” in that it is detecting departure from the null more clearly. More broadly, I am calling any method that is inefficient, incorrect. I believe that the notion that inference based on inefficient use of data is wrong is an important principle of inference.

Perhaps mitigating factors in this case lie in the questions:
1) are the incorrect results produced by the old method clearly incorrect. (That is to say, is there a very real risk that the incorrect results will be unknowingly used or are they so catastrophically wrong that only an MBA student wouldn’t notice)
2) is there a pattern to the datasets that produce incorrect results using the old method and can it be used predictively? If there is no way to practically predict with which datasets the old method will fail, then it is essentially failing randomly - as you suggest.

I suppose that if the answer to either of the above questions is ‘yes’ then it could be argued that your new method should be applied only selectively.

One more argument against dismissing EM because ‘it is not worth the trouble’ is the ability to test against non-standard null values, e.g. noninferiority tests. In this case 1% can easily increase to a more substantial figure.

My question is what if I am NOT interested in detecting the difference, e.g. in case of testing if a new treatment has the same toxicity as a standard competitor. This is partly a question of ethics, but shouldn’t I be more prone to avoid ‘the trouble’ and go for sub-efficient M?