Think of it this way: classically or quantum mechanically, when we pick out a physical system to talk about we are isolating the terms for that system in the universal lagrangian and assuming that in the time of interest that the terms for our physical system couple weakly to the rest of the universe (which happens to contain us).
In principal nothing really weird is going on here and in classical mechanics the idea is totally trivial as far as it goes. On short time scales with appropriately sized actions the deviation from the isolated system and the real system (which is weakly coupled to the world) can be demonstrated to be small as long as the coupling is small.
In quantum mechanics two things complicate this situation. The first is that quantum mechanical systems sort of defy separation into distinct subsets except in special situations. Classically there is a strong sense in which we can point to two different parts of a system and call them separate, but quantum mechanically we really only know how to time evolve _the whole system_ and from a mathematical point of view its the actual object of interest. This is what we are getting at when we talk about entanglement: the two spin 1/2 particles flying away from one another in Bell style experiments are not separate things in the QM description: there is just one wave function.
But in practice I don't think there is any real reason we can't quantize gravity. I'm not an expert but loop quantum gravity seems like a reasonable approach and its very straightforward and its base: just find an acceptable description of geometry and then apply the ordinary quantum mechanical tricks we use to quantize it.
Also, you don't observe things in spacetime. Observations are always purely local. You just infer the existence of spacetime from local observations which are conveniently organized by putting them on a curved 4d Lorentzian signature manifold.
I’ve never quite understood what a quantum theory of gravity would be though. QM involves the observer but gravity engages spacetime - the place where you are observing things. A quantum field theory of gravity seems like a contradiction in terms to me. Unless quantum gravity is really about the Big Bang?