Cleanliness is rather redundant when all your ingredients are granular, and when high temperatures make everything "sticky" (due to fusion or diffusion). Consider...
Say some rust flakes off from the burner or heating element, catches on that pristine white ITC-100 coating, and melts or diffuses into it. Now you've got a black or brown stain. (This is most dramatic in fuel-fired furnaces, where the usually somewhat reducing atmosphere ensures iron is ferrous Fe(II), which is "basic", a flux somewhat stronger than Mg -- often the spot ends up sunken in, as the refractory gets partially melted.)
Incidentally, in pottery: oxides are classed as basic, neutral or acidic. The rules are broadly similar to aqueous chemistry: alkalines are basic (fluxes: Na, K, Ca, Mg..), nonmetals are acidic (B, F, Si, P..), ordinary metals tend to be neutral or basic (Al, Sn..), and transition metals cover the spectrum from basic to acidic, also depending on oxidation state (Fe(III) is sort of neutral; Fe(II) is definitely basic).
A "salt" has a low melting point. CaO happens to have an extremely high melting point, in and of itself, and SiO2, very high; but the combination ~CaSiO3 has a much more modest melting point (liquid at molten iron temps).
Regarding transition metals, they can "salt" themselves even; this is magnetite for example: Fe3O4, better written FeFe2O4, ferrous(II) ferrate(III). Other ferrites are well known for their useful magnetic properties...
The best refractories, in terms of reasonably high operating temperature, corrosion resistance (not just being neutral, but also rather insoluble in a glass melt) and strength, seem to be zirconia based (ZrO2). Pricey though. For example, bricks for lining glass holding tanks cannot flake or dissolve, and have to be in contact with corrosive Na-Ca glass for years. I think they're usually _fused_ from a Zr-Al-Si mixture. An incredibly dense brick like that, has basically no insulation value; they're backed by better insulators (Al-Si firebrick, or poured or rammed cements) to make the thing useful. Wall thicknesses, of course, are in the fractional-meter range, for this reason. Also, glass itself isn't very conductive, so all the more reason to have very good insulation.
Anyway... This, isn't a pursuit for the OCD, or the clean freak, or, uh, the hypochondriac I guess? It can be done in immaculate, contained conditions -- but it's exponentially more expensive to practice that way!
I suppose it doesn't matter much -- I assume most of your questions are hypothetical rather than practical (I mean that you're asking about the possibility, not that you're actually setting up to do it right now -- at least most of the time). So, it's still interesting to explore how much can be put into doing things in the various ways. And then the later question becomes: if/when you want to give it a go, what level of that do you want to attempt, what are you capable of doing, how much budget do you need for materials and such, that sort of thing.
Tim