Sabine Hossenfelder sets out to demystify spin 1/2. That's an admirable objective, and Bee is a good explainer, but I'm not convinced that she succeeds. If we avoid digging into the group theory, probably the simplest way to describe a spin 1/2 system is to say that we need to rotate it through 720 degrees to get it back to the original state.
We can think of macroscopic systems that act sort of like this, and Bee has one. For me, a simpler one is a system of two interlocked gears, one twice the size of the other. Rotating the smaller gear once only turns the big one half-way, while rotating it twice brings everything back to the starting point. The trouble with all such implementations is that it's hard to imagine a supposedly simple elementary particle harboring such complexity in its coupling to spacetime.
Once again, we are forced to realize that, as Feynman put it, "quantum mechanics is not only stranger than you imagine, but stranger than you can imagine." Fortunately, the mathematics is not so limited. If someone wants to understand why there are no, say, 1/3 spin particles, I don't know of any answer except to say "study the math."
But read Bee's article - she has lots of useful stuff to say even if you still wind up somewhat mystified.