I think his argument is that both of these potentials are perpendicular to each other and thus can be decoupled from each other.
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Good point! I had prepared my answer to Smudge and Hakasays similarly, but you were the fastest.

I agree with Smudge about the mistake of equation 4, but we shouldn't throw the baby out with the bathwater.
I don't see the author's text as telling us that charges of the same sign in the outer shell would tend to oppose the attractive force, but that there is stress between them along the surface, which represents energy.
In other words, the repulsive surface tension in a spherical shell of charge cannot be moderated by the presence of another spherical shell of opposite charge near this surface, because the repulsive forces are oriented along the surface, between charges of the same sign close to each other, while the attractive forces by the opposite charges of the inner shell, are oriented perpendicularly to the surface, towards the center.
The attractive and repulsive forces are orthogonal. So to answer Hakasays, I don't think the two forces are competing.
Then the author chose the spherical capacitor because the calculations are simple (even if he was wrong), but his idea is equally valid with a planar capacitor: the charges of same sign would repel each other along the surface, tending to concentrate at the periphery where perhaps they could be recovered.
I will try to clarify the idea with a capacitor that we will charge with only 2 charges on each electrode, see brief diagram attached.
At the beginning these charges represent two dipoles of positive and negative charges.
If they are far apart, each dipole sees the other as two charges reduced to a point, thus as a zero charge, the forces are almost zero.
Charging the capacitor is equivalent to spacing the positive charge from the negative charge of each dipole, i.e. increasing the length of the dipole. When the distance between charges of the same sign starts to be small compared to the length of the dipole, each charge of one sign will "see" the other charge of the same sign and feel the repulsive force since it is no longer neutralized by the charge of the opposite sign.
This repulsive force does not work. Only the energy against the attractive force is to be taken into account for the charge of the capacitor, and yet the repulsive force seems to be able to do some work now, hence the idea of extra-energy.