Anyway, it's now proven by assuming the symmetry of the diameter and not the golden ratio.

Could you elaborate on this symmetry? Not quite sure how it's defined or what it preserves.

Just saying that the centroid is in the middle of the segment between the point of contact and the opposite side of the figure. The golden ratio scaling then appears as a condition for the new centroid to be on the edge (i.e., on the end of the segment.)

Some observations from looking for golden ratio tops:

From a square you can create an infinite number of different tops: any position of the scaled square along the edge would work.

Only stars with an even number of points can be made golden.

Any triangle can be carved in 3 different ways, by placing the scaled triangle on each side in precise locations. But none of the resulting shapes has symmetry (non even for the equilateral triangle).

I tried to make a series of fish eating each other, but it was more difficult than I expected. I'm not very happy with what I have obtained so far:

The cool thing is that all the golden fish tops can be stacked to form a complete fish. Something like a Matryoshka doll.