Can quantum mechanics be more primary than quantum field theory as a fundamental theory? The question is inspired from the Lubos Motl post: http://motls.blogspot.com/2015/05/inconsequential-irrational-incorrect.html which says:

In this approach, quantum mechanics is primary because a Hilbert space
  pre-exists and the field operators have to "adapt" to it. This is very
  different from the approach in which a Hilbert space is produced out
  of local fields, e.g. as the Fock space. In such an approach, the
  spacetime would be primary and fundamental. Such an approach is
  probably inapplicable in quantum gravity.

But we do know that unlike in quantum mechanics where the position operator and thinking about one particle are possible, in QFT, these are not possible. So the two frameworks seem not reconcilable, unless we can somehow embed QFT into quantum mechanics and allow talking about one particle, not just fields.
What exactly is the current state of physics for this matter? Is quantum mechanics more primary/fundamental, or is quantum field theory fundamental? 
 A: The question is akin to asking whether a wooden object or a wooden chair is better for sitting - the chair is a wooden object, and the question nonsensical because wooden chairs are a subset of wooden objects. Likewise, quantum field theory is a quantum mechanical theory. It is simply a quantum mechanical theory with infinitely many degrees of freedom. From the viewpoint of quantization, we are always quantizing a classical Hamiltonian formalism - the quantum field theoretical one happens to be a classical field theory where phase space is therefore infinite-dimensional.
There is nothing in QFT that is fundamentally different from "ordinary" finite-dimensional QM, except that many of the mathematical objects have a problem where they are, strictly speaking, not defined. Some of these problems we know how to repair, some we don't (like the issue of a well-defined path integral for a general QFT).
Also, "QFT" inherently does not need to be relativistic - some of the issues you mention, like the non-existence of a position operator, are relativistic rather than field theoretic in nature, but in condensed matter applications it is perfectly common to have a Galilean QFT.
