Time dependent and time independent Schrödinger equations

The "independent" in "time-independent Schrödinger equation" doesn't mean that the wavefunction $\psi(x,t)$ is independent of time, but that the quantum state it defines doesn't change with time.

Since $\psi(x)$ and $\mathrm{e}^{\mathrm{i}\phi}\psi(x)$ for any $\phi\in\mathbb{R}$ define the same quantum state, this does not imply $\partial_t\psi(x,t) = 0$. Indeed, as the solution shows, the time dependence $\mathrm{e}^{\mathrm{i}Et}$ is precisely the kind of dependence that is allowed.