The Lagrangian and Hamiltonian approaches are frameworks, and not theories. It is certainly true that a wide variety of systems are susceptible to such an approach.
However, there are many theories which do not possess Lagrangians. For example, it is believed that a certain set of six-dimensional superconformal field theories may be able to describe all lower dimensional conformal field theories. However, currently, there is no Lagrangian known for any of them. In addition, in certain conformal field theories, we may use a bootstrap approach to define the theory in terms of the algebra of its operators, the CFT data, without mentioning a Lagrangian.
Furthermore, there is currently no known Lagrangian for M-theory, though there is a matrix model which is believed to be M-theory in a certain limit (see work by Susskind).
Locality and Manifest Unitarity
There is also a known issue emerging with the Lagrangian framework itself, or at least its present implementations. In particular, quantum field theory is normally done in such a way as to make locality and unitarity manifest.
One of the consequences of this is that scattering amplitudes computed in this formalism using Feynman diagrams are much more complicated than their final expressions.
We also cannot currently construct a Lagrangian in a way to make Lorentz invariance manifest without introducing gauge redundancies. Thus, the present formalism is not without its flaws, and there are new formalisms emerging. For example, in the case of graviton scattering, it has been shown that the S-matrix may be constructed by seeing it as offering a representation of the BMS group. This is shown to lead to Weinberg's soft theorems and sub-leading soft theorems.
It would be reasonable to guess that should a 'theory of everything' be constructed, it may not be a standard quantum field theory as we understand it, and points that I mentioned above, especially in the modern literature into scattering amplitudes, hint at this.