It is very difficult for you to consider electron spin as
real spinning.
The main reasons are as follows,
1 The electron size is too small, so by equating the angular momentum of the spinning sphere of the electron to 1/2hbar, the electron sphere speed leads to more than 100 times the speed of light.
2 The spinning electron will not go back to their original forms when they are rotated by an angle of 2 pi.
(This is called "two-valued", so when the angle is 4 pi, they go back.)
To speak simply, Spin is "a magnetic moment", because the magnetic moment can be measured by the real experiment.
But why we call it "spin"? To know that reason, we have to understand the spin history.
The most important phenominon of this was "an anomalous Zeeman effect", I think.
Because it is possible that the Stern-Gerlach experiment and the fine structure (energy difference between 2P1/2 and 2P3/2) can be explained also by the Bohr-Sommerfeld model. (For example, see
this.)
They tried to explain about the many spectrum lines under the magnetic field using the spin-orbital interaction.
(But to be precise, the one electron atom hydrogen usually shows the normal Zeeman effect. and Lithium tend to show Paschen-Back effect. So the anomalous Zeeman effect is a little complicated to explain.)
Sorry for a little long story.