Unfortunately I can't help with the classical picture, but in quantum physics it all comes out very nicely: You can interpret the Lagrangian as giving all possibilities to build a trajectory through spacetime. In the path integral formulation we then follow one such trajectory from one configuration to another configuration and find its amplitude. And then we integrate over all possible trajectories that we could have picked. For incoherent trajectories there will always be another one that cancels out the amplitude. Where the amplitudes add up constructively you will find stationary action and the classical behavior in the limit. So this is a depth-first approach: first follow one trajectory completely, then add up all possible trajectories.

The Hamiltonian approach in contrast is breadth-first: you single out a time axis, start with some initial state, and consider all possibilities that a particle (or field in QFT) could evolve forwards in time just a tiny bit (this is what the Hamiltonian operator does). Then you add up all these possibilities to find the next state, and so you move forwards through time by keeping track of all possible evolutions all at once. This massive superposition of everything that is possible (with corresponding amplitudes) is what you call a state (or wavefunction) and the space that it lives in is the Hilbert (or Fock) space.

So Lagrangian/path-integral: follow full trajectories, then add up all possible choices. depth-first

Hamiltonian/time-evolution: add up all choices for a tiny step in time, then simply do more steps: breadth-first

I imagine it a bit like a scanline algorithm calculating an image as it moves down the screen (Hamiltonian) vs something like a stochastic raytracer that can start with an empty image and refine it pixel by pixel by shooting more rays (Lagrangian)

This is my layman explanation anyways...hopefully it helps, even though i can't say much about their relationship in classical physics.