I don’t think it is that simple, in cartesian coordinates, integrating by parts should give a more complicated expression. It should not really change anything in the end, the final expression e.g. in post #7 says essentially that the difference between two integrals is some number, and since a coordinate change should not change the values of the integrals, that should hold in cartesian coordinates as well.Fine, but do you agree with me that p4p4p^4 is Hermitian in Cartesian coordinates for the hydrogen atom wave functions? Technically, when you do a partial integration in Cartesian coordinates, you only need the boundary values at x,y,z→±∞x,y,z→±∞x,y,z\rightarrow\pm\infty, not at r→0r→0r\rightarrow 0 as in spherical coordinates. Since everything behaves well at infinity, all the boundary terms vanish in Cartesian coordinates so the operator is Hermitian.
The problem with spherical coordinates is that it puts a boundary at the place where it should not really be, that is at r→0
This might introduce additional artificial singularities, but not doing it should not change anything about the singularities actually present in the physics problem (e.g. the Coulomb potential is singular at zero, no matter what coordinates you use).