There is nothing wrong, the point of confusion as you noted lies in the characteristic of the ground ring $R = \mathbf{Z}[X]$. When $R$ has characteristic $0$, then the ghost map sending $W(R) \to R^\mathbf{N}$ is invertible whence an isomorphism. If you want or need more details on the ghost map, recall that $W_n(\underline{x}) = \sum_{i=0}^n p^i x_i^{p^{n-i}}$ is the $n$-th Witt polynomial, and the bizarre Witt sum and multiplication polynomials arise by insisting that the function $W(R) \to R^{\mathbf{N}}$ which sends $\underline{x}$ to $(W_0(\underline{x}), W_1(\underline{x}), \ldots)$ is a ring homomorphism, where we give $R^{\mathbf{N}}$ the usual component-wise operations. The key point is that when $p$ is a unit in $R$, then this map is also a bijection which can be seen by recursively unwinding the Witt polynomials. So the weird sum formula you have is the result of summing in an isomorphic copy of the usual component wise addition and then truncating.

In any event, you shouldn't expect 'usual' the formulation of "multiplication by $p$" in the ring of $p$-typical Witt vectors when the ground ring is {\it not} of characteristic $p$.

Note that the ghost map is completely useless if $p = 0$ in $R$. This is why to derive the sum and product formulas, one first works in a $\mathbf{Q}$-algebra, to get a universally valid formulation which then may be applied in any characteristic. If you want a great introduction to this, see Jacobson's "Basic Algebra II", Section 8.10, page 497 or Serre's classic "Corps Locaux"/"Local Fields".