I've only had a quick look at the Coddens paper, but if I understand it correctly what he is doing is the equivalent of saying that the set of unit vectors are not vectors, because they don't form a vector space. You can't 'add' two unit vectors together and always get another unit vector.

He mentions Hestenes and geometric algebra in his introduction, which in my view gives a much clearer intuitive geometric picture of spinors. (I'm not clear on why he rejects this.) In geometric algebra, the spinors are just the even sub-algebra - the linear combinations of all products of an even number of basis vectors. Since each basis vector represents an (oriented) reflection, pairs of basis vectors represent rotations (i.e. pairs of reflections).

In 2D, the geometric algebra consists of linear combinations of a scalar (1), two basis vectors (x and y) and a bivector representing the plane (xy). The even sub-algebra is the linear combinations of $1, xy$ where $(xy)^2=-1$. The 2D spinors are thus just the complex numbers.

In 3D, the geometric algebra consists of a scalar (1), three basis vectors (x, y, z), three bivectors representing coordinate planes (yz, xz, xy), and a trivector (xyz) representing the oriented volume element. The even sub-algebra is the linear combinations of $1, yz, xz, xy$ where $(yz)^2=(xz)^2=(xy)^2=-1$. The 3D spinors are thus just the quaternions.

If we have two generic vectors $a$ and $b$, then $ab=a\cdot b+a\wedge b=|a||b|(\textrm{cos }\theta+B \textrm{sin }\theta)$ where $B$ is the unit bivector in the plane of $a$ and $b$ and $\theta$ is the angle between them. If the vectors being multiplied are parallel, the result is a pure scalar, if perpendicular then the result is a pure bivector. The dot product is familiar from vector algebra, and is just the scalar part of the geometric product. The wedge product is the bivector part, and is dual to vector algebra's cross product. (We have to use the dual to turn it into another vector because vector algebra can't cope with bivectors. However, this doesn't work perfectly because it transforms wrongly under reflections - i.e. the cross product gives a 'pseudovector', not a vector. Pseudovectors are actually bivectors.) Thus both products from vector algebra are unified in a single product in geometric algebra, and correspond to finding the 'real' and 'imaginary' components of a spinor.

The complex column vector representation of spinors arises from noting that $xy$ acts like the imaginary unit $i$, so $\alpha(1)+\beta(xy)=\alpha+\beta i$ and $\gamma(yz)+\delta(xz)=(\gamma+\delta(xy))(yz)=(\gamma+\delta i)(yz)$ so the quaternion can be encoded as two complex coefficients of a vector space over the basis $1, yz$.

In geometric algebra, spinors implement rotation by conjugation. To rotate a vector $v$ by the rotation represented by the spinor $S$, we find $SvS^{-1}$. If $S$ has a norm different from 1, then $S^{-1}$ has the reciprocal norm, and they cancel. This is why a (non-zero) scalar multiple of a spinor corresponds to the same rotation, and why we can add spinors together without giving rise to any problems. We commonly use unit spinors (called rotors) to represent rotations so that we have an unambiguous representation, like we commonly use unit vectors to represent things like surface normals, where the length doesn't matter. But using unit spinors to represent rotations doesn't mean they're not elements of a vector space any more than using unit vectors does. It's also the case that the sign cancels too, so $-S$ gives the same rotation as $S$, but is not the same spinor.

I find that geometrically, the best way of visualising a 3D spinor is as the angle between a pair of oriented reflection planes. ('Oriented' means the plane has a well-defined front and back.) If the planes are identical, the reflections cancel, and the identity results. (i.e. with parallel vectors we get a pure scalar 1.) As you rotate one plane with respect to the other, the results is a rotation through twice the angle between the planes, about the axis along which the planes intersect. When the angle between the planes reaches 180 degrees, the planes are parallel, but with normals in opposing directions. Again, the reflections cancel, so this represents a 360 degree rotation, but this is not the same spinor! The 180 degree angle means one plane is the negative of the other. If you keep on rotating the reflection planes, eventually a 360 degree angle between reflections corresponds to a 720 degree rotation, and we have identity again. It is not true that rotating a spinor 360 degrees gives the opposite sign, and you have to rotate 720 degrees to return to where you started. What happens is that rotating the spinor through an angle increases the corresponding rotation by twice the angle, so rotating a spinor 180 degrees negates the spinor but turns the rotation right around, and rotating the spinor 360 degrees corresponds to a rotation of 720 degrees. The spinor is not the rotation - it is the angle between the pair of (oriented) reflection planes that make it up.

So spinors do have a perfectly understandable geometric interpretation.