[jessriedel]

I'm mostly unable. It was musing about these questions that got me interested in really understanding the Legendre transform.

I can say that dynamical equations that try to generate spatial translation from initial data on a time-like slice, rather than time translation from a space-like slice like Hamiltonian dynamics, are doomed because there is no well-posed initial-value problem (except in certain special cases involving massless particles), e.g., you generically cannot infer what's far from a spatial plane even if you know everything that happens on that plane for all time. Related topics:

http://www.scholarpedia.org/article/Hyperbolic_dynamics https://en.wikipedia.org/wiki/Well-posed_problem

Also, I would say that the fact that the Legendre transform is a manifest involution gives (quite) weak evidence that there are no other major formulations to find. Of course, it's possible to use a hybrid strategy, Routhian mechanics, with Lagrangian and Hamiltonian formulations on different degrees of freedom:

https://en.wikipedia.org/wiki/Routhian_mechanics