Is it mathematically as well physically alright (consistent) to not quantize metric when doing QFT in curved spacetime? I am not well-versed with the theory of renormalization of QFT as of now (I just know that when you redefine your observables you get rid of the UV divergences and then get finite predictions from your theory) so this question might not make any sense.
When we do QFT in curved spacetime as described in sec $2.1$ of Parker and Toms monograph: we replace

*

*$\partial$ by $\nabla$

*$\eta_{\mu \nu}$ by $g_{\mu\nu}$

*$d^nx$ by $|g|^{1/2}d^nx$
I am feeling a bit uneasy when I look at this minimal coupling prescription (in fact it seems ad-hoc as well) because when I am doing QFT any field which is dynamic on spacetime is quantized either by defining a commutation relation or using path integral. The background here as in the language of QFT in curved spacetime is simply $\eta = diag(1,-1,-1,-1)$ it doesn't vary with $x$ coordinate as $g_{\mu\nu}$ does.
So why are we justified to not quantize the metric which may in fact be dynamic as in the case of collapsing part of a collapsing star? Here dynamic might not be the right word because to define dynamic nature of something we need ruler and watch to see the change or mark the event which is given by the metric. It's kind of circular reasoning here. And what's more mysterious to me is getting prediction out of this theory.
Maybe this mysticism and uneasiness is stemming out of my illiteracy of renormalization so can someone kindly explain to me why not quantizing the metric is a right choice till we probe curvature of order $(\frac{1}{l_p})^2$ where $l_{p}$ is Planck's length.
There is a similar question as well it doesn't justify why leaving metric un-quantized (classical) is a good approximation though my question is intimately connected to backreaction which is discussed there. I want to know a bit more explicitly why working with the classical metric is okay on physical as well as mathematical grounds.
 A: I don't know the monograph of "Parker and Toms", but from the title of your question it seems that it deals with "QFT in curved space-time". In this field those effects are studied that come into focus by considering a non-dynamical non-Minkowskian metric. This is already much more complicated than standard QFT in a Minkowskian metric.
Of course one can go one step further and also "quantize the metric". However, this is no longer "QFT in curved space-time", that would be Quantum Gravity.
And Quantum Gravity is even more complicated, up to date there is no unique established theory accepted by the physics community. I think, the simple reason why "Parker and Toms" don't quantize the metric is that it is out of scope of their monograph.
For a treatment of the "quantisation of the metric" one has to read a book on Quantum Gravity, or even better several books as there are several theories (string theory, quantum loop gravity etc.) that are proposed. And probably one would not simply "quantize the metric", but other more complicated and abstract objects somehow related with the metric.
EDIT: When it comes to predictions, QFT in curved space-time already makes predictions which can be checked. There are effects (Hawking radiation for instance) where the "full theory" (quantisation of space-time) is not necessary. It is similar to the General Relativity where the deviation of light by massive objects can already predicted by the linear approximation of the field equations whereas the full theory is only  necessary to compute correctly Mercury's perihelion motion. Physics already showed in its history that simple models often achieved very good descriptions without including the full complexity of the dynamics.
