To calculate scattering amplitudes, we consider $$ A(i\to f) = \langle{f | \hat S|i} \rangle = \langle{f |\mathrm{e}^{ -\frac{i}{\hbar} \int_{-\infty}^{\infty} dt' H_{\mathrm{i}}(t')} |i}\rangle$$ $$ \equiv \langle{f(\infty) |\mathrm{e}^{ -\frac{i}{\hbar} \int_{-\infty}^{\infty} dt' H_{\mathrm{i}}(t')} |i(-\infty)}\rangle$$ where all objects are given in the interacting picture and $H_i$ denotes the normal-ordered interaction Hamiltonian in the interaction picture. (The Hamiltonian is normal-ordered to get rid of all self-loop diagrams.)
Moreover, the asymptotic in and out state are defined in terms creation operators, e.g. $$ a^\dagger(k)a^\dagger(k') |0\rangle \equiv |k,k'\rangle \equiv |i\rangle , $$ where $|0\rangle$ denotes the ground state of the free theory. (For the sake of argument, we can imagine that the experiment happens in an isolated box. We prepare the particles outside of the box, send them into the box where they interact, and afterwards observe them outside of the box again. Moreover, we isolate the two incoming (or outgoing) particles sufficiently such that we can assume that no interactions happens and we can treat them non-interacting particles. In other words, they are eigenstates of the free theory, i.e. $H_0$.)
By using these formulas it's possible to calculate cross sections without ever mentioning something like an interacting vacuum $|\Omega\rangle$. (This is done, for example, in Tong's lecture notes and the book by Lancaster and Blundell.)
On the other hand, many books emphasize that the interacting vacuum $|\Omega\rangle$ is essential and then discuss the usual ways to handle it (LSZ, Gell-Mann low).
Where exactly and how does $|\Omega\rangle$ enter the story? What goes wrong if we calculate cross sections in the "naive" approach discussed above?
PS: Tong seems to suggest on page 75 that the distinction between $|\Omega\rangle$ and $|0\rangle$ only becomes important once we consider more sophisticated questions such as how to
"compute the viscosity of the quark gluon plasma, or the optical conductivity in a tentative model of strange metals, or figure out the non-Gaussianity of density perturbations arising in the CMB from novel models of inflation."
To answer these questions we must evaluate correlation functions of the form $$\langle \Omega | \phi_H^1 \ldots \phi_H^n |\Omega\rangle$$ which involve the interacting ground state $|\Omega\rangle$ (and Heisenberg picture fields). The LSZ formula then allows us to evaluate these expressions by relating them to the scattering amplitudes we are familiar with that involve the free vacuum $|0\rangle$ (see Eq. 3.95 in Tong's notes).
However, I've never seen a standard qft textbook that treats these more complicated questions. Thus it would be strange that most textbooks emphasize the importance of the LSZ formula if it were only relevant for these advanced questions. On the other hand, the thing on the left-hand side in the LSZ formula (involving the interacting vacuum $|\Omega\rangle$ and Heisenberg fields usually comes seemingly out of nowhere.)
