First lets dispel the myths. Fluxing does not allow the tin, antimony and lead to mix properly, or prevent their separation. Tin and antimony alloy (mix) with lead easily, and once alloyed, neither can be separated from lead except by sophisticated methods avilable to laboratories. Fluxing has no role in that.
Fluxing cleans the alloy, but only in the respect that it separates the dirt and dross from the alloy and allows them to be skimmed off the top of the melt. It does not get the dirt out of the alloy. Basically, the dirt and dross float to the top being lighter than the alloy, but that requires vigorous stirring. Dirt is what's on the lead before melting, dross is created by oxidation. So they are different, but removal is the same so they can be regarded as the same for practical purposes.
Although the alloy has been cleaned before use, it will accumulate dross over time. The hotter the alloy, the more dross it will accumulate. It takes the form of a thick sludge on the top of the melt. The sludge is particles of dirt and dross entrapped in the alloy. If you don't mind losing some alloy you can just skim off the sludge.
The only purpose of fluxing is only to save the alloy by separating the dross out. There can be a lot of it in a fresh batch of alloy. On one occasion I weighed five pounds of it from a 60 pound melt. If it is not fluxed out, a lot of alloy can be lost if the sludge is skimmed off without fluxing.
I note your concern about flaming. Some books say that flaming is not desirable, that it retards the effect, but I have found that it works better if it flames. Note that you don't need fancy fluxes. The foundries consider them too expensive for the volumes they melt, and use the dry drossing technique, which is sawdust and caustic soda.
Here's how I do it. I use cheap common candles as a flux. Cut them into one inch lengths. Drop one piece into the melt and stir it in. Wear a welding glove to protect hand from the flames. After a few seconds stirring it should flame. If it doesn't, the second piece should do it. Keep stirring as it burns. It will burn for a few minutes. You should end up with a pile of fine powder-like dust. If it is more solid than powdery, it has not fully separated. Give it another go. Finally, skim it off.
Nothing I write is theoretical, it is all tried and tested. I use this fluxing method for every 60 pound bulk ingot melt from raw materials, and whenever the sludge on my casting melt gets thick enough to remove.