Numerical calculation of a quantum field's observables
If you're looking for a formulation of QFT that resembles Schrödinger's equation in single-particle QM and that can be solved on a (infinitely fast) computer, here it is:
Scalar fields
For a single scalar field with Hamiltonian $$ \newcommand{\pl}{\partial} H \sim \int d^3x \Big[\big(\dot\phi(x)\big)^2 + \big(\nabla\phi(x)\big)^2 + V\big(\phi(x)\big)\big], \tag{1} $$ simply replace continuous space with a finite lattice, treat $\phi(x)$ as a collection of independent real variables (one for each lattice site $x$), and use $$ \dot\phi(x)\propto \frac{\pl}{\pl \phi(x)} \tag{2} $$ normalized so that the (lattice version of the) canonical commutation relation holds. Then the equation of motion in the Schrödinger picture is $$ i\frac{\pl}{\pl t}\Psi[\phi,t] = H\Psi[\phi,t] \tag{3} $$ where the complex-valued state-function $\Psi$ depends on time $t$ and on all of the field variables $\phi(x)$, one for each lattice site.
Gauge fields
The lattice Schròdinger-functional formulation for gauge fields associates elements of the Lie group (not the Lie algebra!) with each link of the lattice (pair of neighboring sites). The analog of the differential operator (2), corresponding to the time-derivative of a gauge field, is nicely described in section 3.3 of https://arxiv.org/abs/1810.05338. The state-function $\Psi$ is a function of all of these group-valued link variables, and gauge invariance is expressed by a Gauss-law constraint.
Fermion fields
The lattice Schròdinger-functional formulation for fermion fields is conceptually the easiest of all, because the anticommutativity of fermion fields (Pauli exclusion principle) means that you only have a finite number of possible values associated with each lattice site (instead of, say, a continuous real variable like in the scalar-field case), so the Schròdiger equation (3) is just a gigantic matrix equation in this case. In practice, it's messy.
Some complications
Of course, there are a few complications:
Solving a partial differential equation (3) with a ga-jillion independent variables takes an awful lot of computer power, far more than we currently have, unless we settle for compromises like using a lattice with only a handful of sites in each dimension.
As far as I know, we don't yet know quite how to put chiral non-abelian gauge theories on a lattice. In particular, we don't yet know quite how to put the Standard Model on a lattice. (X-G Wen has suggested that we actually do know how to do it, at least in principle, but I haven't seen it spelled out yet in terms that I understand.) But we know how to put QCD and QED on a lattice, and it's done routinely, subject to the limitation noted in the first bullet above.
Figuring out which states $\Psi$ represent single-particle states is a difficult problem, not to mention the multi-particle states that you'd need to do scattering calculations or to study bound-state properties (like hydrogen in quantum electrodynamics). There are QFTs in which its easy, like non-relativistic QFTs and trivial relativistic QFTs, but it's surprisingly difficult in relativistic theories like quantum electrodynamics where an electron is always accompanied by an electric field and an arbitrary number of arbitrarily low-energy photons. This difficulty is related to the fact that observables in QFT are tied to regions of spacetime, not to particles. Particles are phenomena that the theory predicts, not inputs to the theory's definition.
For more detail, I recommend the book Quantum Fields on a Lattice. There are several books about lattice QFT, but this is one of the most comprehensive.
Strictly non-relativistic QFT
Things are easier in strictly non-relativistic QFT, like the example shown in the question. In that case, the number of particles is fixed, and the model decomposes into separate non-relativistic QM for each fixed number of particles. Each one of these separate non-relativistic models has far fewer independent variables, proportional to the number of particles instead of proportional to the number of points in space.
Yes, QFT books are unfortunately often vague about the connection to reality until the chapters where they discuss scattering, in my experience. The measurement-related postulates in QFT are the same as in QM but with some difficulties. $| \langle p | \psi \rangle |^2 $ is the probability density of getting a momentum $p$ in an experiment measuring momentum; this is used in the derivation of the scattering matrix. From that, it follows that $\langle \psi | P^\mu | \psi \rangle$ is the expectation value of momentum in a given state, for example. Whether you then put the time dependence in $|\psi \rangle, P^\mu$, or both is up to the picture being used (schrödinger/heisenberg/interaction).
One big difference to QM is that position space doesn't seem to have a satisfactory description in QFT (no lorentz invariant position space state for >1 particle unless you have multi-time arguments). It is often argued that a position-space description is not necessary, with something vague like "no localization possiblie in QFT". I happen to not believe that is true, but I warn you that my view is not the majority view in that regard.