Does spacetime position not form a four-vector?

You are correct.

Position is a vector when you are working in a vector space, since, well, it is a vector space. Even then, if you use a nonlinear coordinate system, the coordinates of a point expressed in that coordinate system will not behave as a vector, since a nonlinear coordinate system is basically a nonlinear map from the vector space to $\mathbb{R}^n$, and nonlinear maps do not preserve the linear structure.

On a manifold, there is no sense in attempting to "vectorize" points. A point is a point, an element of the manifold, a vector is a vector, element of a tangent space at a point. Of course you can map points into $n$-tuples, that is part of the definition of a topological manifold, but there is no reason why the inverse of this map should carry the linear structure over to the manifold.

And now, for a purely personal opinion: While Carroll's book is really good, the physicist's way of attempting to categorize everything by "transformation properties" is extremely counterproductive, and leads to such misunderstandings as you have overcome here. If one learns proper manifold theory, this is clear from the start...

Great reasoning: as in Uldreth's fantastic answer but I would add one more thing that may help cement your good understanding in place.

Co-ordinates are absolutely not vectors, they are labels on charts and are no more vectors than your street address is a vector. Almost certainly the reason people make the implication that you have correctly identified as wrong is this: in flat space (i.e. Euclidean, Minkowski or generally signatured spaces), affine co-ordinates for positions can have two roles: they are both labels and (once one has chosen an origin) superposition weights that combine linear basis tangents to the Euclidean (Minkowski ...) manifold linearly to yield a general tangent to the manifold. If you think about it, what I have just said is a slightly different take on Uldreth's second paragraph that begins "Position is a vector ...".

It's worth saying that I definitely recall the following learning sequence as a teenager. When beginning high school at about age 11, I was first shown co-ordinates (Cartesian of course) as labels. I suspect that this is how they are introduced to all children. I distinctly recall the idea that only two years later was the notion (that only works for Cartesian and generally affine co-ordinates) of a point's co-ordinates as a position vector introduced. Before that I had a very clear idea of a vector as a displacement or link between two points, an idea that, through the appropriate limit, leads to the tangent idea in a general manifold. On reading your question, I laugh when I recall the teacher's implying that the second role of co-ordinates as position vectors was a "new and advanced" way to look at vectors, whereas on the contrary it is a way of thinking that you correctly understand to be very limited and only workable in the affine case.

Here is a bare bones easy way to see that coordinate tuples are not 4-vectors.

Start in an inertial coordinate system in flat spacetime. Change the coordinate system with a constant translation:
$x' = x + A $
$y' = y$
$z' = z$
$t' = t$

Even in this idealistic case, 4-vectors and coordinate tuples transform differently. The components of the 4-vectors don't change at all in this case, while the coordinate tuples do.