While the following may be less “physically” intuitive than the question intends, I liked it when I heard it. If you believe in the importance of Poisson brackets, it is natural to ask if they make the Lie algebra of some transformation group, or in other words, if they can be realized as Lie brackets of vector fields.
This question was answered in the 1960s by “prequantization” of symplectic manifolds, but in the case of $X=\smash{\mathbf R^2}$ with points $x=(p,q)$ and 2-form $\omega=dp\wedge dq$, already by Sophus Lie in (1890, p. 270): one considers $L=X\times\mathbf U(1)$ with points $ξ=(x,z)$, projection $ξ\mapsto x$ and connection (contact) 1-form $\varpi = p\,dq + dz/iz$. Then an automorphism, $g\in\operatorname{Aut}(L,\varpi)$, is a diffeomorphism of the form $$ g(x,z) = (s(x),ze^{iS(x)}) \tag1 $$ where $s\in\operatorname{Aut}(X,\omega)$ is a symplectomorphism and the function $S$ is determined up to an additive constant1) by $$ p\,dq-s^*(p\,dq)=dS. \tag2 $$ Now the Lie algebra $\operatorname{aut}(L,\varpi)$ is isomorphic to $(C^∞(X), \{\cdot,\cdot\})$: to any $\varpi$-preserving vector field $Z$ we can attach the function $H(x) = \varpi(Z(ξ))$ called its Hamiltonian, and conversely any $H ∈ C^∞(X)$ gives rise to the infinitesimal automorphism $$ Z(x,z)= (\operatorname{drag}H(x),iz\ell(x)) \tag3 $$ where $\operatorname{drag}H=\smash{\bigl(-\frac{\partial H}{\partial q},\frac{\partial H}{\partial p}\bigr)}$ is the symplectic gradient, and $\ell= p\smash{\dfrac{\partial H}{\partial p}}-H$ is the Lagrangian for you.
Paths $t\mapsto x$ in $X$ “lift” to paths in $L$ where $z$ spins around the “internal clock” $\mathbf U(1)$ so eloquently described in Feynman’s QED, evolving with $S=\int\ell\,dt$. “Wave functions” live in the complex plane it spans, and constructive interference of the waves corresponds to stationary $S$.
Added: At a more classical level, I can only recommend reading the 18-page Introduction of Souriau (1997, French version) where he explains from first Newtonian (or really D’Alembert) principles, not the Poincaré-Cartan 1-form $p\,dq - H\,dt=(p\dot q-H)dt=\ell\,dt$ but the Lagrange 2-form $dp\wedge dq - dH\wedge dt$ and generalizations which are (he argues) the only intrinsic objects worth talking about at non-quantum level. At (12.98) he explains why he abandoned the variational approach of a previous book — just as Lagrange also did 200 years earlier. Worth pondering?
1) That is to say, one has a central extension $1\longrightarrow\mathbf U(1)\longrightarrow\operatorname{Aut}(L,\varpi)\longrightarrow\operatorname{Aut}(X,\omega)\longrightarrow 1$ which “integrates” $0\longrightarrow\mathbf R\longrightarrow C^∞(X)\overset{\operatorname{drag}}{\longrightarrow}\operatorname{aut}(X,\omega)\longrightarrow 0$.