In fact the Schrödinger equation is a postulate so it can hardly qualify as an explanation, if I follow your argumentation. For me, though, it gives an explanation (why does the Schrödinger equation give you an explanation while the Bohr model does not? Both are just theories...).
Just a few words more
The Schrödinger equation is a postulate ...up to a certain extent
The actual postulate is the duality particle / wave
Once you accept this postulate, the equation can be derived by merging the D'Alambert equation ( wave) and the Hamilton equation (particles) via Debroglie
The similarity between rays of light and material particles had already been noticed in the 18° century but nobody dared to draw the logical conclusion
Apart from that the Schrödinger equation can be applied also to the study of the atom of hydrogen but it is not, so to speak, "Hydrogen " dependent
On the contrary
The postulate
m*v * 2PI * r = nh
It has been proposed by Bohr "ad hoc " for the atom of hydrogen due to the fact that Bohr knew in advance the results to be achieved
For this reason the Schrödinger equation is a step forth in the study of the secret of atoms
