Position of protons and neutrons in a nucleus

This does not violate the exclusion principle because the exclusion principle merely states that there cannot be more than one fermion in the same quantum mechanical state. In the case of two protons and two neutrons, the different particle species don't exclude each other to begin with (because a neutron state is different from a proton state).

Furthermore, that they have the same expectation value for position doesn't mean that they are in the same state. States can coincide with their expectation values for some observables but not for others. In this specific case, the states likely differ by their spin (one proton/neutron has "spin up" and the other "spin down").

Pauli's exclusion principle states that two fermions can't occupy the exact same quantum state simultaneously. Two fermions can have spatial wavefunctions that overlap with nonzero values at common locations. That is fine - the point is that the entire spatial wavefunctions (along with spin states) can't be the same for both particles.

Thus, all four particles are most likely found in exactly the same space, at the central point.

This doesn't seem to say that the particles overlap entirely. This sates that wave functions of all the particles are centered around a common central point. Pauli's principle doesn't forbid that.