Why is Schrodinger equation taught while it does not describe an electron? Strictly speaking, it is "wrong" because it does not describe spin-1/2 particle like an electrons. Why in every QM textbook is it taught, not as a historical equation, but as a current equation?
 A: 
Why is Schrodinger equation taught while it does not describe an electron?

This question is of the same order as asking:
"Why is Newtonian gravity still taught everywhere since it has been seen  that General Relativity is the underlying theory of gravity"  and thus should explain everything?
There are even observations in the solar system.
Physics is the observation and of nature and the specific use of measurements to record observations.Mathematical theories are used in order to model observations and , important, predict new observations so that the theory is validated.
In physics mathematical theories, there is a definite process of emergence, classical emerges from quantum, thermodynamics from statistical mechanics etc. Each mathematical format of equations  is used in the appropriate range of variables.
Schrodinger's equation modeled  the concept of quantum mechanics which , historically was developing, and fitted the hydrogen spectra in a consistent way. Fitted them within errors in measurement at the time. This is a fact and useful. As the observations increased and the errors decreased the necessity of spin for the fine structure of the spectra became necessary. This does not invalidate the previous solutions, it adds details in fine structure finding that the effect of the spin of the electron can be detected in the spectra .
So the Shrodinger equation is still used because its solutions are  a first order fit to quantum mechanical observations, spins adding  a small measurable effect.
For the same reason the harmonic oscillation  model is widely used  because it is a first order approximation to quantum mechanical potentials about their minimum.  Even though it is a simple model it is still very useful in disciplines that need quantum mechanics to model their data.
A: I think you are confusing the treatment of relativistic (spin-$\frac{1}{2}$ particles) electrons as compared to the non-relativistic case. The Schrodinger equation can perfectly describe the properties of the non-relativistic electron. The Dirac equation describes the interactions of relativistic electrons (and other spin-$\frac{1}{2}$ particles).
