Question

What are problematic issues of quantum field theory in curved spacetime, when accepting semiclassical limitation?

Answer 1

As far as my reading goes, the literature seems to suggest that for a fairly broad class of background manifolds, a "time function" exists on all points of a manifold

I am not sure if I will expand this into an answer, but the biggest problem with a classical background is quantum information is lost in a way similar to what happens in measurement. The metric back reaction is a classical physics system responding to a quantum event. The black hole is not fully quantum mechanical and this does not treat the problem in the way we think of quantum states of a hydrogen atom emitting quanta of photons.

Answer 2

The first point is uncontroversial and expected to be solved fully by quantum field theory in curved space-time (QFTCST).