The Hartree-Fock approximation in solid-state physics, if I understand correctly, is the assumption that the many-particle wavefunction can be written as a Slater determinant of single-particle wavefunctions. This makes sense to me in the contexts in which Hartree-Fock approximations are discussed. However, I am having difficulty reconciling this with my notion of second quantization. The ways in which I've seen second quantization introduced (see here, for example) starts with the initial assumption of a single-particle Hilbert space. A Hilbert space with $n$ particles is obtained by $n$ tensor products of the single-particle Hilbert space, boson/fermions are defined as subspaces of these spaces which have the appropriate symmetrization/anti-symmetrization, and creation/annihilation operators take you from the $n$ particle Hilbert spaces to $n \pm 1$ particle Hilbert spaces (or alternatively, they move you around in a much larger Fock space). In this definition, the wavefunction of $n$ fermions appears to be defined with a Slater determinant in the exact same way that the Hartree-Fock approximation does. This makes sense to some degree - with second quantization, one often talks about creating/annihilating individual particles, so there had better be a notion of a "single particle." But I always had the impression that the second-quantization way of thinking about things was more fundamental - it was just how we were defining systems with multiple particles. Does this mean that every time we use second-quantized notation, we're implicitly making a non-trivial Hartree-Fock-like assumption about our system? This seems to morally conflict with how I think about QFT, where we have no issues tossing around creation/annihilation operators and talking about particles without ever worrying about such an approximation.
There's no contradiction here. The Fock construction is just telling us that the entire space can be spanned by states that are Slater determinants. It doesn't imply that a given state has that form; that's exactly the additional assumption that the Hartree-Fock approximation makes.
As an example, consider the state $$|\psi_1 \rangle \otimes_A |\psi_2 \rangle + |\psi_3 \rangle \otimes_A |\psi_4 \rangle$$ where the $|\psi_i \rangle$ are orthogonal single-particle states and $\otimes_A$ is the antisymmetric tensor product. Each of the individual terms is a Slater determinant, but it's easy to check that the superposition isn't, i.e. that there aren't any coefficients so that $(\sum_i a_i |\psi_i \rangle) \otimes_A (\sum_i b_i |\psi_i \rangle)$ yields that state.
Your mistake is thinking that any state in second quantization is a Slater determinant. This is not true, you can form linear combinations of such states. More generally, obviously, any state in the Fock space can be written in "second quantization formalism". I say obviously because in a sense this is how the Fock space is built.