The problem is not with the tools you use. the problem is with your lack of understand of how pixels work. They are square. And the worst part is what you're trying to accomplish will be used for a computer display, which is basically a buildup of pixels.
So no matter how much you try to blur an image, jaggedness will still be present. And please do not ever dare resize a 16x16 to 32x32. That's just crazy.
Here's my tip: I know that it is frustrating when you can't accomplish what you want to do, but please stop complaining. Do your research not on graphics editing but on icon making. Study some icons that are circular. Even those iPhone types which has rounded corners. Do you notice how they manage to make those curves smooth? That subtle shadow. Optical illusion is the answer to your problem. Duplicate your image, desaturate, adjust opacity. That's the simplest way. And that's the only thing you have to blur. Good luck.