The question is inspired from the answer to Is a QFT in a classical curved spacetime background a self-consistent theory? (I am going to reference this link as "The Q" to avoid confusion).
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 (http://www.theorie.physik.uni-goettingen.de/forschung/qft/research/theses/diss/Solveen.pdf page 10 and possibly the answer to "The Q" also and others). If this is indeed the case - corrct me if it is not - then would not the Point 3 of the answer to "The Q" addressed, as we now have time? (And also the Point 2, as this seems to take care of in-out states issues.) Or is there some problem that would not allow us to use ordinary normal ordering QFT Fock space machinery/tools?
In the comment to the answer to "The Q", there was a talk about defining QFT in a curved spacetime without reference to Minkowski vacuum. I do not understand what this is supposed to mean. Do these approaches eliminate the concept of "vacuum"? (For all observers, the tangent space to the point they are at is Minkowski spacetime, so I guess this is the case?)
For this alone, inspired from the last slide in http://www.damtp.cam.ac.uk/user/pz229/Research_files/QFTCS.pdf : What is the last slide exactly saying? This slide does not seem to talk about renormalization issues, and it seems to talk about quantum vacuum technically having infinite energy. But don't we ignore this issue anyway when we decided to define a QFT in a classical background spacetime? That is, doesn't this divergence issue go away once we fix a classical background spacetime (of course this is, in many ways, not a correct way of doing physics, but we do this as an approximation anyway). And what would be other divergence issues in QFT on a curved spacetime (specifically restricted to issues introduced by defining QFT on a curved spacetime)?
QFT in a curved spacetime does suffer from having a fixed classical background spacetime - but let us for now forget about the issues caused by this, including back-reaction problems. Other than the problems listed above, what will be other issues in QFT in a curved spacetime?