That seems to be a misunderstanding. Velocity-dependent/generalized potentials $U(q,\dot{q},t)$ are allowed in Lagrangian mechanics. See e.g. this, this, this & this related Phys.SE posts.

1. That seems to be a misunderstanding. Velocity-dependent/generalized potentials $U(q,\dot{q},t)$ are allowed in Lagrangian mechanics. See e.g. this, this, this & this related Phys.SE posts.

2. It should be stressed that a generalized force $Q_j$ might not have a potential, not even a velocity-dependent/generalized potential. Example: Dissipative forces.