In terms of the code on the Phoible website, what you see is what you get, and it looks to me as though there isn't a built-in facility for asking open-ended questions like you have in mind. You would have write appropriate code for processing the database, though there could be useful utilities already available.
Practical usefulness is hard to quantify. A classical database question of theory is whether any language contrasts "pharyngealization" and "rounding", or could they be subsumed under one feature – they were reduced to one feature in the Fant & Halle system. Unawareness of such contrasts allowed a simpler feature system, but turns out to have been empirically wrong.
An example of an unanswered question that might benefit from such a tool connected to the Phoible database is, does any language clearly contrast [ʉ] with [ɯ]? I manually determined that the answer is "no (not clear)", but there are about 4 languages where the claim is at least worth looking at in greater depth. Processing the indicia of reliability would also be useful, since many people (erroneously) think of specific languages as being well-defined objects whose properties are self-evident. One of the candidate languages is "Khams Tibetan", which is more a language family rather than a single language. For various reasons (available from the database), I consider the claim for that language to be "not clearly established" (multiple languages, multiple opinions i.e. lack of consensus, unpublished sources).
There is a tradition in linguistics of presenting sweeping conclusions about human language based on ill-defined criteria and unretrievable connections of actual language data. To the extent that one could devise and test well-defined formulas stating a claim X, and could say something systematic about how strong – or weak – the claim is, that could be a positive contribution to phonological research.