We are indeed "fielded questions like this a hundred times over." That's why I try to write some papers to allay it: It never works. Anyway, here's one example that's relevant for your queries: https://scirate.com/arxiv/1601.04360. My own view is that taking first-person elements seriously within physics (as QBism does) *adds* to the notion of reality, rather than taking away from it or ignoring it (as full-blown instrumentalism does). Also QBism retains an element of "something like" a structural realism that is usually ignored (namely, in the structure of the Born rule as a normative condition that's good for anybody). These things are discussed in the paper just cited. I hope that helps. Maybe Ruediger will want to add something.
Best wishes, Chris
1. It was just a goofy thing that I thought would get the readers to smile. But Wolfgang Pauli did have quite a mystical interest in 137 precisely because of its connection to the fine structure constant. This is documented in quite a number of places; the book by Suzanne Gieser, "The Innermost Kernel: Depth Psychology and Quantum Physics. Wolfgang Pauli's Dialogue with C. G. Jung," is quite a good source. I just quoted from something I could easily get my hands on at my desk. Preceding the quote I placed in the paper, Enz wrote, "The depth of Pauli's philosophical views incited Arthur Koestler to remark that Pauli `perhaps had a deeper knowledge of the limits of the natural sciences than most of his colleagues'. One of these limits which disturbed Pauli during all his scientific life was the duality between the electric field strength and the means to measure it by its action on a charge ... The disturbing aspect is that the precision of such a measurement is limited by the atomicity of electric charges as given by Sommerfeld's fine structure constant 1/137, which has not yet been explained."
2. That one might *think* or *get the impression* that QBism will be destroyed as an interpretation of quantum mechanics if the SICs do not exist in every finite dimension is a perennial worry of my colleagues Ruediger Schack and David Mermin. They don't want people to get that impression. They feel I place too much emphasis on the mathematical problem since, as an interpretation of QM, QBism need not depend on any technical innovations in the formalism. And they are correct: QBism, as a self-consistent interpretation, neither stands nor falls on the existence of SICs.
What the SICs would add is just a very pretty way of rewriting the Born Rule to be purely in terms of probabilities, and with a pretty way of expressing the idea, the hope is that it will facilitate philosophical conversations. For instance, the Oxford philosopher Harvey Brown wrote in a recent paper, "Powerful plausibility arguments have long been available, some since the birth of QM, to the effect that the quantum state is something real. They almost all have to do, in one way or another, with quantum phase, with the fact that the wavefunction, in its relation to probability, is strictly a (generally complex) probability amplitude: it has more structure than a probability distribution does." Well, it is just wrong that a quantum state has more structure than a probability distribution does. And a SIC representation of quantum states (if it exists) helps the Born Rule, written in terms of probabilities, look as simple as possible. The hope is that if one can make it look so pretty and simple, even a philosopher might take note. Probably fat chance, but I haven't given up yet.
As far as whether one needs the SICs even for that conceptual discussion, surely one doesn't. One can already make the conceptual point with a representation based on any minimal informationally complete POVM consisting of rank-1 elements. Schack and I make that point in arXiv:1412.4211 [quant-ph]. On the other hand, if one wants to take the Born Rule in probabilistic terms as a *fundamental axiom* of quantum theory, then I have the feeling that such an axiomatization will be most facilitated if the SICs exist: For, it would allow the Born Rule---as an axiom---to *sneak in* so much of the structure of quantum theory with one simple statement, it might just be God's hammer. Some baby steps in that direction can be found here arXiv:1612.03234 [quant-ph].
I hope that clarifies things for you.
With best wishes, Chris