I've discovered that, when I've got the foreground color set to a color other than white or black--ANY color--I get the saturation capability I'm looking for. Why that should work I have no idea, but it seems to.
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Actually, not ANY color - greys will work just the same as white or black. The reason is simple: white, black or greys have zero saturation. That's why painting with them in "Saturation" mode changes saturation toward zero. If you want to increase the saturation, you should paint with some very saturated color - saturated red, or green, or blue, or something else - hue doesn't matter, only the saturating does. If the saturation of your brush color is smaller than that of source image, it will desaturate. If it's bigger - it will increase saturation. No magic, just some easy math.