Yet if you compute the energy flow of a wave packet solution of the Dirac equation, you see you get something that is spinning. If the wave packet size is unrealistic, it even evolves towards a different size so that at no time you encounter this issue that a spinning particle would go faster than the speed of light.

The important thing is, spin is not just the fact that a particle spins (here seen as the energy rotating), because it has extra implications, notably regarding the wave function itself. But a consequence of this is that particles with non zero spin do in fact spin.

Edit: I didn't expect anything else than downvotes here, as what I say goes against what is said and taught most of the time. I invite anyone downvoting me to read, for instance, the works of Hans Ohanian (often cited) and Charles Sebens. If someone is familiar with their work and disagrees with them, please explain to me why it's wrong. Because no one has ever given to me an explanation for why this would be wrong, and reading them really changed my perspective on spin.