What is the physical content of the principle of least action?

OP's question seems to be essentially a version of the inverse problem for Lagrangian mechanics, i.e. given a set of EOM$^1$ $$E_i(t)~\approx~ 0,\tag{1}$$ does there exist (or not) an action $S[q]$ such that the EOM (1) are the Euler-Lagrange (EL) equations $$\frac{\delta S}{\delta q^i(t)}~\approx~ 0,\tag{2}$$ possibly after rearrangements? This is in general an open problem. See however Douglas' theorem and the Helmholtz conditions mentioned on the Wikipedia page.

Physically, in the affirmative case, there is a functional Maxwell relation $$\frac{\delta E_i(t)}{\delta q^j(t^{\prime})}~=~\frac{\delta E_j(t^{\prime})}{\delta q^i(t)}. \tag{3}$$ See also e.g. this related Phys.SE post.

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$^1$ The $\approx$ symbol means an on-shell equality, i.e. equality modulo EOM.