*Edit:* The result is fine: Hopf's lemma was proved in

* G. Giraud, *Problèmes de valeurs à la frontière relatifs à certaines donn ás discontinues*, Bull. de la Soc. Math. de France, 61 (1933), 1–54

Below is my incorrect answer (which I keep for reference), where I mistakenly assumed that $C^{1,Dini}$ is stronger than $C^{1,\alpha}$.

***

*Old answer:*

Looks like you are right: the regularity assumption on the boundary is insufficient, although I did not check this very carefully.

You may have a look at the paper *A counterexample to the Hopf-Oleinik lemma (elliptic case)* by D. E. Apushkinskaya and A. I. Nazarov, [DOI:10.2140/apde.2016.9.439](https://doi.org/10.2140/apde.2016.9.439), [arXiv:1503.02179](https://arxiv.org/abs/1503.02179). Let me quote from p. 2 of this paper:

> The reduction of the assumptions on the boundary of $\Omega$ up to $C^{1,Dini}$-regularity was realized for various elliptic operators in the papers [Wid67], [Him70] and [Lie85] (see also [Saf08]). A weakened form of the Hopf-Oleinik lemma (the existence of a boundary point $x_1$ in any neighborhood of $x_0$ and a direction $\ell$ such that $\frac{\partial u}{\partial \ell}(x_1) \ne 0$) was proved in [Nad83] for a much wider
class of domains including all Lipschitz ones. We mention also the paper
[Swe97] where the behavior of superharmonic functions near the boundary
of 2-dimensional domains with corners is described in terms of the main
eigenfunction of the Dirichlet Laplacian.
>
> The sharpness of some requirements was confirmed by corresponding
counterexamples constructed in [Wid67], [Him70], [KH75], [Saf08], [ABM$^+$11] and [Naz12]. In particular, the counterexamples from [Wid67], [Him70] and
[Saf08] show that the Hopf-Oleinik result fails for domains lying entirely in
non-Dini paraboloids.

Later, the paper shows that lack of "Dini condition" on the boundary invalidates the assertion of Hopf's lemma.