How can the hydrogen ground state have zero angular momentum if the Bohr radius is nonzero?

In the Bohr model, the electron in the $n=1$ state moves in a circular orbit of radius $a_0$ around the proton. Bohr's angular-momentum quantization condition gives

$L=mvr=\hbar.$

Thus, the Bohr model predicts nonzero orbital angular momentum for the ground state.

However, the quantum-mechanical ground state of hydrogen has zero orbital angular momentum and is spherically symmetric.

This appears paradoxical: if the electron has a characteristic separation $a_0$ from the proton, why does it not have orbital angular momentum?

PS: Why does the Bohr model nevertheless give the correct hydrogen energy levels, despite predicting non-zero $L=\hbar$ for the ground state rather than $L = 0$?