I'm quite happy to admit that I don't know the classical-quantum CQ programme well, and certainly not well enough to hazard a genuinely informed opinion that might take the form "it's a candidate for a fundamental theory" or "it's a candidate for a better EFT". I think I can say it isn't obviously not one of those, though. Also, I'm not ready to sloganize the work beyond "it's complicated".
"... invariant under spatial diffeomorphisms ... consistent ... completely positive, norm preserving, and linear in the density matrix ... the metric remains classical even when back-reacted upon by quantum fields ... the dynamics here, while stochastic, can leave the quantum state pure -- it is rather the classical degrees of freedom, which gain entropy."
It's a heavy couple dozen pages (part V is a mountain for mountaineers (I am nowhere near the summit) and the foothills are also hard work, though most of it shouldn't pose technical comprehensibility problems anyone who's done QFT (GKSL is prominent [short intro <https://arxiv.org/abs/1906.04478>], and the Lindbladian is the generator of time-translations) and who has encountered Lagrangian and Hamiltonian formulations of gravitation, especially if they've glanced at Birrell & Davies. A crash read or refresher of Wald's QFTCS https://arxiv.org/abs/gr-qc/9509057 might be useful if like me you've mostly spent time in one of the two fundamental silos and want a bit more meat than what's in the first couple pages of Oppenheim 2018's part I. The general relativistic constraint equations are also important, and there is a good overview at https://link.springer.com/article/10.1007/s41114-020-00030-z
The meat, for me, is the potential for a better approximation than semiclassical gravity when quantum fluctuations are large: "[even if not fundamental, CQ gives] the ability to consistently study back-reaction effects in cosmology and black-hole evaporation ... [but] care should be taken, since an effective theory might violate our assumptions of Markovianity or complete positivity at short time scales or when the gravitational degrees of freedom have not fully decohered.", "This theory serves as a sandbox in which to understand issues around the quantisation of the gravitational field. After all a probabilty density \rho and the Liouville equation have a lot in common with the wave function \psi and the Heisenberg equations of motion. [and the rest of that paragraph just before (eqn 2)]".
A good entry point to CQ is Oppenheim's [hep-th] https://arxiv.org/abs/1811.03116
"... invariant under spatial diffeomorphisms ... consistent ... completely positive, norm preserving, and linear in the density matrix ... the metric remains classical even when back-reacted upon by quantum fields ... the dynamics here, while stochastic, can leave the quantum state pure -- it is rather the classical degrees of freedom, which gain entropy."
It's a heavy couple dozen pages (part V is a mountain for mountaineers (I am nowhere near the summit) and the foothills are also hard work, though most of it shouldn't pose technical comprehensibility problems anyone who's done QFT (GKSL is prominent [short intro <https://arxiv.org/abs/1906.04478>], and the Lindbladian is the generator of time-translations) and who has encountered Lagrangian and Hamiltonian formulations of gravitation, especially if they've glanced at Birrell & Davies. A crash read or refresher of Wald's QFTCS https://arxiv.org/abs/gr-qc/9509057 might be useful if like me you've mostly spent time in one of the two fundamental silos and want a bit more meat than what's in the first couple pages of Oppenheim 2018's part I. The general relativistic constraint equations are also important, and there is a good overview at https://link.springer.com/article/10.1007/s41114-020-00030-z
The meat, for me, is the potential for a better approximation than semiclassical gravity when quantum fluctuations are large: "[even if not fundamental, CQ gives] the ability to consistently study back-reaction effects in cosmology and black-hole evaporation ... [but] care should be taken, since an effective theory might violate our assumptions of Markovianity or complete positivity at short time scales or when the gravitational degrees of freedom have not fully decohered.", "This theory serves as a sandbox in which to understand issues around the quantisation of the gravitational field. After all a probabilty density \rho and the Liouville equation have a lot in common with the wave function \psi and the Heisenberg equations of motion. [and the rest of that paragraph just before (eqn 2)]".