In particular, the different stability features of attractive and repulsive Coulomb forces does not seem to come into play.
In general, with experience from fluid dynamics, one may say that attraction is connected to acceleration and as such is stable, while repulsion is connected to retardation and so is unstable.
An illustration is given by a pendulum in usual hanging position (attraction) and inverted upright position (repulsion), as illustrated in the above picture. We see converging attractive forces (in red) balancing gravitation (green) giving a stable hanging position and diverging repulsive forces giving an unstable inverted upright position.
It is natural to connect to the Helium atom with two electrons in opposition by repulsion as studied in this sequence of posts, which may be unstable by the above mechanics and thus allow a variety of non-symmetric configurations with one of the electrons dominating, which possibly could be the origin of the mysterious concept of electron spin.
The stability of electron configurations is rarely studied in quantum mechanics, since the focus is put on eigenvalues and energy levels and not on often multiple eigenfunctions.
SvaraRaderaThis is very far from the truth. If you look into the field of electronic structure calculations, some enlightenment on the subject should be gained.
In particular, the different stability features of attractive and repulsive Coulomb forces does not seem to come into play.
The Coulomb interaction is part of the electron correlation energy. Martin's book Electronic Structure could be of some initial help to you.