Yes, all true, but we have the Hamiltonian formulation (e.g. https://arxiv.org/abs/1505.01403 https://arxiv.org/abs/1505.01403 which is a nice book chapter) whi

Yes, all true, but we have the Hamiltonian formulation (e.g. https://arxiv.org/abs/1505.01403 https://arxiv.org/abs/1505.01403 which is a nice book chapter) which “… is also the starting point for the canonical quantization program, which constitutes one main approach to the as yet unsolved problems of Quantum Gravity. In this approach one tries to make essential use of the Hamiltonian structure of the classical theory in formulating the corresponding quantum theory.” From there you could jump to https://en.wikipedia.org/wiki/Canonical_quantum_gravity https://en.wikipedia.org/wiki/Canonical_quantum_gravity which has references.