Question:Why might a flat-bottom finite well be a better approximation of the potential well confining nucleons than a Coulomb well tapering to a lowest energy in the middle

The strong force is much stronger than the Coulomb force within the nucleus.

The finite potential well (also known as the finite square well) is a concept from quantum mechanics.

A particle is contained in a "box" with finite potential "walls" as an extension of the infinite potential well.

The potential confining the nucleons have contribution from the strong interaction attractions between all nucleons and the coulomb repulsion between all pairs of protons.

The strong force is much stronger than the Coulomb force within the nucleus.

Thus, the potential is more like a fiat bottom well, which is a good approximation of the strong interaction attraction potential when nucleons are close, than a Coulomb well.

In fact, the nucleons would not keep together if only Coulomb force exists.

The repulsion will cause all the protons to fly apart.