A better understanding of the screening effect and how it works?

What I am taught-

Inner shell electrons shield the outermost electrons from the attraction of nucleus. More closer the inner shell is to the nucleus, more its shielding effect. thus $$n_s > n_p > n_d > n_f$$ and such (in terms of screening effect as "s" orbitals are closest to the nucleus). $$Z_{\text{eff}} = Z - \sigma$$; where $$\sigma$$ is the screening constant and can be calculated via Slater's rule.

Why I think this happens -

Nucleus attracts the outermost electron just as well according to coulomb's formulae ; but the inner shell electrons provide an repulsive force towards the outermost electrons and thus "decrease" the attractive forces. But I don't get how $$n_s > n_p > n_d > n_f$$ is working; like s is closer to the nucleus thus is more far away from the outermost electrons thus it should provide less repulsive forces, right ? But what I think this order is due to is the shapes of these orbitals; like the p orbitals are like dumbells due to which some repulsive forces may get cancelled out (like derivation of electric dipoles and stuff); but I don't think that shows the full picture.

Adding to it; I know about the penetrating power of the orbitals ; but doesn't it just plot a graph of $$4 \pi^2 \psi_r^2$$ as a function of $$r$$ (distance from nucleus) of different orbitals and seeing the graph we can predict about the distance of the orbitals from nucleus; and we can deduct that "s" orbital stays the closest and the distance increases from there on. But it again pulls the question why then s orbital has highest screening effect, because if its closest to the nucleus it should be farthest from outer shell electrons; thus should have least screening.

• As the $1s$ electrons are nearly against a nucleus with $Z$ protons, the outer electrons feelsthe attraction of $Z$ protons minus the $2$ electronic charges due to these $1s$ electrons. If these $1$s electrons were more cumbersome, and distributed around a large domain around the atom, the outer electrons will "feel" the attraction of the $Z$ protons minus something smaller than $2$ Oct 10, 2021 at 14:55
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– uhoh
Oct 10, 2021 at 23:51
• Screening is direct application of classical electrostatics. Inner spherical charge distribution gives full screening like if the charge were in a nucleus. Deviations from spherical symmetry and reaching outer regions behind the screened electrons decrease the screening. Therefore, overlapping of 2 orbitals negatively affects screening by the more inner orbital. Oct 11, 2021 at 5:29