Strictly speaking, quantum field theory is still quantum mechanics. For systems involving a fixed number of particles, you can still try to keep track of the dynamics of each one of these particles by defining $\hat{x}_i,\hat{p}_i$ for all of them. Then you can write down the Hamiltonian $H(\hat{x}_i,\hat{p}_i)=\sum_i \frac{\hat{p}_i^2}{2m}+V(\hat{x}_1,...,\hat{x}_N)$, where the $V$ could be potential or/and particle-particle interactions. Then you can try to solve for the spectrum and eigenstates, which is in fact notoriously difficult to do. Even for a single particle, we are only able to write down the exactly solution for a handful of cases (e.g. infinite potential well, harmonic trap etc.).
The advantages of QFT include, but not limited to: (1) We can forget about the detailed dynamics of each individual particle, and focus on the creation and annihilation of this type of particle (since they are all identical if we only have one species) at each spacetime point using $\hat{\psi}^\dagger(t,x),\hat{\psi}(t,x)$. It's like looking at the waves on a lake instead of tracking down each water molecule. (2) When special relativity is incorporated, we have relativistic QFT, which can deal with cases where particles can pop up from vacuum as long as you give it enough energy ($E=mc^2$). (3) Perturbation theory can also be very systematically done using Feynman diagrams.
To summarize, using $x$ as an operator or label, there is no contradiction. It's a matter of convenience.
Edit:
Based on comments below from Tobias and Prox. I realized that in writing the above answer I overlooked the subtlety of defining position operator in relativistic quantum theories. I also found another well-explained post here along the same line with the comments.