In his introduction to Electrodynamics, Griffiths wants us to prove what the electrostatic field of a dipole is like in a (he says) "coordinate-free form". For convenience let's focus on proving the formula for spherical coordinates, independently of the position of the origin. This is purely a geometrical problem.
Let $\bf{p}$ be the vector corresponding to the dipole moment, and $\bf{r}$ the position vector determining the direction of the spherical unit vectors. Griffiths, citing this figure,
tells us that $\pmb{p} = p\cos \theta\ \pmb{\hat{r}} - p\sin \theta\ \pmb{\hat{\theta}}$, which is the key point of his argument, but to my eyes this is true only when $\pmb{p}$ is colinear to $\pmb{\hat{z}}$. In the rest of the cases, we should have $$\pmb{p} = p\cos \alpha\ \pmb{\hat{r}} - p\sin \alpha\ \pmb{\hat{\theta}} + p\cos \beta\ \pmb{\hat{\phi}}$$ with $\alpha := \angle \pmb{p\hat{r}}$ and $\beta := \angle \pmb{p\hat{\phi}}$.
Now the problem is that his reasoning fails with this last characterization of $\pmb{p}$. Am I wrong, or is his argument not coordinate-free? If I'm wrong, why?
I assume that you’re referring to the solution of Problem 3.36 (p. $160$ here) as provided on p. $73$ here.
You’re right that this part of the derivation makes reference to a particular coordinate system and is thus itself not coordinate-free.
The task is not to derive the coordinate-free form without using coordinates. The task is only to show that the electric field can be written in a coordinate-free form. The solution does this by expressing the solution derived using spherical coordinates in a coordinate-free form, using only products and sums whose values are independent of the coordinate system. Since the field has a unique value, if it can be derived in a particular coordinate system and the result expressed in coordinate-free form, then this result is valid without reference to any coordinate system.