There is more than one way to answer the question of 'what probability is'; you are advocating for an inductive approach where the general category is arrived at through exposure to particular instances; another approach is to work within a more general framework which expresses different types of probability theories as configurations of more general types. Granted, one must start inductively in order to get the general framework I speak of—but such frameworks are now in place, so it is not necessary for everyone who wants to compare theories of probability to proceed inductively. Then again, if your expertise is philosophy of probability you should do it anyway—but that brings me to the other point I wanted to make, which is: in my original post I was not referring to philosophers of probability, since that would be their expertise; instead I was referring to ancillary knowledge and whether someone acquires foundational or 'philosophical' (i.e. meta) knowledge of these ancillary fields.
As for whether there can be value to knowing about something like QM without knowing the math behind it, consider Feynman's preface to Q.E.D (the edition he wrote for the layman [well, transcription of lectures technically]). In it he describes the six years of training required in order to actually perform the mystical rites of physics, but knowing what the rites are about and what purpose they serve etc. is also valuable, which is why he wrote the book/gave the lectures (sorry, trying to give a rough quick summary, but the analogy he presents in the preface is the best explanation I've seen).