The problem I have with larger sizes is if you don't add flux, there's just not enough flux that gets into the joint for the amount of solder you need.
Really? I think this might be brand specific. I'm sure larger solder wires can have "enough" flux in them.
Sounds like large voids are the likely culprit IMHO, and solder quality definitely has an effect here.
But keep in mind, that there's different amounts of flux available for a given alloy and flux type. There's 1.1% (also referred to as P1 or Core 50), 2.2% (P2 or Core 58), and 3.3% (P3 or Core 66) by weight for rosin types of flux for example. The smaller amounts will leave less residue (cosmetically pleasing when skipping cleaning), but is really only suited to clean parts and boards to start with given there's so little flux.
Does multi-core solder not fix this problem?
Multicore is meant to reduce the size of voids in each flux core. And the smaller the void, the smaller the splatter. FWIW, I only find it an issue with larger diameter solder wire, such as .062" (mine pops like hell). I wouldn't worry about it with smaller diameters.
Another method is to feed solder into a solder wire cutter/perforator (cuts a slit/punches holes down the length of the wire so the gas can escape without "blowing out" molten solder & flux). FWIW, Hakko offers such units, and there are copies available.
I've never tried Multicore (I think that's Ersin, right?). It doesn't seem to be particularly popular in the US, where Kester rules, but I seem to see it popping up elsewhere.
Multicore was an Ersin trademark (US company), but it's currently owned by Henkel of Germany. They've shifted manufacturing from the US to other locations, such as Malaysia, and it's become a more European brand (better availability there than in the US, though you can still get it). Stannol is another quality brand that's rare in the US.
Multicore has also reduced their lead based offerings, while Kester, AIM, and Indium haven't. Combine this with better availability, these other brands have a larger market share IME (Kester having the largest piece).