The trouble im stuck with is that after I extract, edit and Nurb the control points of the circle, the new geometry is not a circle seen from the top view

What you extracted are points lying on the circle, not its control points.

Here’s a good introduction to how Bezier/ B-spline curves work if you are not so familiar with them.

Perfect circles are a little more complicated because the control points are weighted (rational) ,but you can approximate them as higher degree non-weighted curves using the Rebuild curves component

Tried to rebuild and the resulting geometry is inaccurate ( I dont know why because it worked like a charm with the circle )
Tried to take the points as they are and I suppose the knot type is wrong because im getting pointy vertices

The trick with the last solution was assuming the rebuilt control points were ordered in a nice and symmetrical way - I got it to work by adjusting the seam point to align with the X axis:

And also, using sift-pattern in that way was a little bit of a hack to “select” control points which were ordered in a regular manner, and is of course specific to the degree and number of points used to rebuild the curve. If you’re working with more complex shapes, try looking into attractors to locally influence CPs instead ( lots of examples here, just search around)