Is there a way to stop Adobe Illustrator from adding anchors when using the Pathfinder or Shape Builder?

When using the Effects > Stylize > Round Corners... Illustrator option on shapes I built using Pathinfder and Shape Builder, I'm having a distinct problem with doouble anchor points. Since the effect does not round corners if they consist of multiple anchors, this spoils the effect at multiple places.

Take this shape, for example. The shape is fully expanded and consists of only a single path.

I built this using Pathfinder and Shape Builder. Look what happens when I round this shape's corners:

At the weirdest places, the Builder or the Pathfinder have created shapes using multiple anchors for a single point.

Is this adding of points preventable? If not, is it possible to remove them with a function? Object > Simplify does not seem to work, as it keeps the additional points even when not 100% accurately reproducing the shape.

The methods mentioned in this question don't seem to work for the Shape Builder.

I'm working on cs6.