quantum field theory - How does QFT account for localization (into a finite volume of space), in practice?

In Quantum Field Theory there is until yet no agreement (as far as I know) on the issue of localization of particles. When one talks about a 'particle' in QFT, one usually means a single-particle state of definite momentum, or a wavepacket made out of such states. It is not clear, however, what (if any) are the states that correspond to something that is localized in space, or even something that is localized into a finite region of space.

Some textbooks on QFT (e.g. Peskin and Schroeder, page 24) suggest that (at least in the case of the free Klein-Gordon theory) the field operator $\phi(\vec{x})$ creates a particle at position $\vec{x}$, i.e., the state \begin{equation} |\vec{x}\rangle := \phi(\vec{x})|0\rangle \end{equation} would correspond to a particle localized at $\vec{x}$. However, it can be easily shown that such states are not mutually orthogonal, i.e., $\langle \vec{y}|\vec{x}\rangle\neq 0$ if $\vec{y}\neq \vec{x}$. So these states cannot possibly correspond to localized particles.

This bothers me and I would gladly hear other people's views on this. Still, I can imagine, for instance, that these states do actually correspond to effectively localized states, by which I mean that in practice it makes sense to regard them as localized states, even if they technically aren't. But this is only a shot in the dark; I have no idea whether that makes any sense. And if this is the case, then what is the justification for this view?

Other references advocate that one should use the eigenstates of the so-called Newton-Wigner position operator, which is explained in detail in this excellent answer. Although these states also have their peculiarities, they seem to be preferable over the states $\phi(\vec{x})|0\rangle$.

So theoretically it is not clear how we should describe localized particles. Nevertheless, in collider experiments, for instance, the particles (or perhaps I should say the quantum fields) clearly are effectively localized into a finite region of space. And there the theory really works! So apparently we are able to describe localized particles. So how does one describe this spatial dependence, in practice? I imagine one uses some kind of wavepackets? And does this give any insight into the theoretical problem?