From the OP:one consistent thing throughout all the editions (and through most fiction I can think of quickly) is that for relatively basic items (of which the WoSDD would be one) once you know what they do, that's it - you know what they do.
Which means, the owner/user of a WoSDD is highly likely to know - or easily be able to learn - its general degree of reliability; and then be able to reasonably expect that to reflect in the fiction going forward.
This objection/complaint rests on assumptions that don't have to be accepted (ie they are not essential for RPGing).
First, it assumes that using the Wand is not itself a type of check that can be resolved in a Character => Situation => Setting fashion. Once we abandon that assumption, we can see that the relationship between using the Wand at time 1, and then making a manual search at time 2, is simply a special instance of the general rules question When are retries allowed?