Geometrically exact shell kinematics are governed by mid-surface displacement and rotation of the normal material line. Neglecting through-thickness deformation, the normal line is modeled as a director. Unlike Euclidean vectors, the director lies on the unit sphere, complicating kinematics and dynamic time integration. This study presents an intrinsic time integration scheme for geometrically exact shell dynamics that enforces the director's spherical constraint exactly. By ``intrinsic'', it means that the method evolves the director directly on the unit sphere, without relying on the time evolution of rotation parameters or the complex relationships between their time derivatives and the directors' angular velocities involving tangent tensors. By revisiting the kinematics of directors, the two independent components of angular velocity and the corresponding rotational variations are rigorously identified. A linear relation is derived between the time derivatives of these independent angular velocity components and their associated rotation vector components. This relation is incorporated into the generalized-$\alpha$ scheme, naturally reflecting the geometric constraints of directors, in contrast to the three-component formulation used in conventional rotations. Drawing inspiration from Lie-group integration of Br\"uls and Cardona~\cite{Bruels2010} and intrinsic integrator schemes by Bauchau et al.\cite{Bauchau2013}, a generalized-$\alpha$ scheme specifically tailored for directors is developed. The shell formulation adopts a three-field Hu–Washizu variational principle~\cite{Gruttmann2006}, in which independent stress resultants and sectional strains are introduced. This framework yields element-level equations that allow elimination of the independent variables. Inertial forces are derived from Hamilton’s variational principle, with interpolation applied directly to the mid-surface velocities and the two independent components of the directors' angular velocities. The proposed scheme is validated through numerical examples of increasing complexity. The results demonstrate the second-order accuracy of the method and show lower errors compared with the time integration schemes reported for the numerical examples investigated.