(1) First argument
An ordinary object that is spinning on an axis has an angular momentum which is determined by how the mass of the object is distributed about the axis, and how fast the object is spinning. For a fixed angular momentum, if the mass is distributed farther from the axis the angular velocity is lower; if the mass is distributed closer to the axis, the angular velocity is higher. Think of a spinning ice skater who turns at one, slower speed with arms extended and at another, higher speed with arms pulled in overhead (on the axis).
No size has been found for electrons; they appear to be point particles. If an electron has finite size (which happens to be too small to see), it introduces issues in any classical description, like charge self-repulsion having infinite energy or having a surface which spins faster than the speed of light. If the electron is a point particle, i.e., has no finite size, then it cannot have angular momentum due to spinning about it's own center of mass because the entire object is on its rotational axis. How to get out of this conundrum?
You cannot "see" an electron to determine whether it is spinning or not. The "spinning" of the electron is not measurable; it makes no sense to speak of it in science. However, you can measure an electron's angular momentum; it makes sense to speak of angular momentum in science. Therefore, don't think of the electron as a "spinning" object (which we can never know or observe); think of it as simply having "intrinsic" angular momentum.
(2) Second argument
The usefulness of an analogy in science is determined by whether you can make inferences or understand other features in a first system under study by way of the analogy and knowledge of a second system.
Thinking of spin as "spinning" introduces conceptual problems classically, does not generalize easily to massless objects, doesn't handle half-integer spin well (the representations of a classical spinning object do not look like a spin 1/2 object), and allows you to infer almost nothing correctly about the electron, other than feeling like you know where the angular momentum comes from. This point of view doesn't work well to start with, and is a dead end as you keep studying physics.
On the other hand, thinking of spin as intrinsic angular momentum avoids all the above-identified issues and, you will find with further study, fits nicely into the rest of physics.
(3) Regarding spin-1/2 and the value ℏ√3/2, these come from group theory and a particular choice of units for angular momentum. This can't be really explained well in a post; study group theory for physicists.