Having read through some recent posts regarding the Lagrangian formulation being interpreted into an initial value problem rather than the familiar boundary condition problem we are familiar with, I have a question regarding Hamilton-Jacobi theory (HJT).

My understanding of (HJT) is that we can view canonical transformations as an infinitesimal advancement in time using Hamilton's principle function and therefore regard a problem as an initial value problem much like Newtonian physics. This is supported by the following result from many classical physics texts, where the Hamilton-Jacobi action differs from the Lagrangian action up to some constant $S_0(\boldsymbol{q_0})$ which disappears upon taking variations $\delta S$. \begin{equation} S(\boldsymbol{q,\alpha},t)=\int ^{t}_{t_0}\mathcal L(\boldsymbol{q,\dot q,}t)dt+S_0(\boldsymbol {q_0}) \end{equation} Where $\alpha$ is any constant of the motion, for instance the initial momentum conditions $\alpha _i =p_i(t_0)$. This suggests that the HJ action is the lagrangian action + some other term, sort of like an initial value problem advancing the action step by step.

Firstly is the above true? Secondly if so, how come Hamiltonian mechanics can be thought of as an initial value problem but Lagrangian cannot?