Answer
Electric repulsion and magnetic forces pose problems.
Work Step by Step
A beam of ionized atoms would spontaneously spread apart, making it difficult to focus them down to a tight beam.
Moreover, as the charged ions passed through the inhomogeneous magnetic field inside the Stern-Gerlach apparatus, the magnetic Lorentz forces would dominate the tiny forces due to electron spin. This would make the resulting data very hard to interpret.
