If you think the answer is 'yes of course' let's see you do it for the finite square well!
It's automatic only in the following sense: you have a bunch of states and can always define stepping operators that take you from one state to the next. But as a practical matter, the fact that the hydrogen atom can be solved by the use of stepping operators is far from automatic: it's a miracle!
The Hamiltonian is a quadratic expression in x and p, and the miracle is that this expression can be factored into a product of linear factors. You can explain this as do to the remarkable algebraic properties of Hermite polynomials and Laguerre functions. Or you can ascribe it to 'extra' symmetries, such as the Runge-Lenz vector. But there are very few systems that can be handled this way.