If $n,\alpha>1$ then for every integer $k$ the congruence class of $kp^{\alpha-1}$ is a solutions to the congruence $$x^n\equiv0\pmod{p^{\alpha}},$$ so there are at least $p$ solutions. In particular the number of solutions is not bounded by any function of $n$.

If $n,\alpha>1$ then for every integer $k$ the congruence class of $kp^{\alpha-1}$ is a solution to the congruence $$x^n\equiv0\pmod{p^{\alpha}},$$ so there are at least $p$ solutions. In particular the number of solutions is not bounded by any function of $n$.

Your argument fails because it fails to consider the differentiability condition of Hensel's lemma: The lemma requires that $$(X_0')^n\equiv R\pmod{p^{\alpha-1}} \qquad\text{ and }\qquad n(X_0')^{n-1}\not\equiv0\pmod{p}.$$ So the lemma, and hence your argument, works only if both $n$ and $R$ are coprime to $p$.