Without context, OALD's "illustrative example" is rather odd, since on average we're probably more used to being advised to be more introspective and self-analytical than we would be naturally, when dealing with situations that need "resolving".
But in fact, it's from a book by policeman-turned-author-turned-social anthropologist Dr. Malcolm Young, in the context of police officers often needing to act quickly and dispassionately in emotionally-charged situations (where they should just "follow procedure", and not get emotionally or intellectually involved).
In general, the construction X should be done with the minimum Y is used it contexts where the speaker (or writer) implicitly acknowledges that some Y will be necessary, but it should be kept to a minimum.
Of course, it's always possible X can be achieved with no Y at all. But if the speaker is an authority on the subject, you should assume this isn't likely. The advice isn't to avoid Y completely (which would probably be phrased differently). You're being advised that you will need Y, but as little as possible.