Let me preface this by saying that I don't have an issue with this:
$$ \langle\Omega|T\phi_H\cdots\phi_H|\Omega\rangle = \frac{\langle 0|T\phi_I\cdots\phi_IS|0\rangle}{\langle 0|S|0 \rangle}, $$
what I want to know is why we want to calculate $\langle\Omega|T\phi_H\cdots\phi_H|\Omega\rangle$ at all.
Say you have an initial state $|i\rangle$ in the interaction picture of a real scalar field theory which is an eigenstate of the free Hamiltonian but not of the interacting Hamiltonian. However, just because this isn't an eigenstate of the Hamiltonian doesn't mean it's unphysical, it's just a superposition state:
$$ |i\rangle = |\Omega\rangle\langle\Omega|i\rangle + \sum_{n=1}^{\infty}|n\rangle\langle n|i\rangle $$
where $|n\rangle$ are the eigenstates of the full interacting Hamiltonian (whatever they might be). You can expand $|i\rangle$ in terms of a product of fields acting on $|0\rangle$ (the vacuum of the free theory):
$$ |i\rangle = \prod_{a=1}^{b}\int\left(d^3y^{(a)}\ 2E_{q^{(a)}}\ e^{iq^{(a)} \cdot y^{(a)}}\phi[t;y^{(a)}]\right)|0\rangle $$
and there's still nothing wrong with this statement. If you're really having consternations you can expand $|0\rangle$ in terms of $|\Omega\rangle$ and $|n\rangle$ and get everything in terms of the interacting Hamiltonian eigenbasis (if you do this and then evolve forward in time you end up with the first equation). You can do a similar construction for $|f\rangle$, the final state.
Then what you want to calculate is the probability that $|i\rangle$ has evolved into $|f\rangle$ after some time $t_1 - t_0$ has passed:
$$ P = \langle f|U_{int}(t_1,t_0)|i\rangle $$
and then using the above expansion for $|i\rangle$ and $|f\rangle$ you can get:
$$ P = \\ \prod_{a=1}^{b}\int\left(d^3y^{(a)}\ 2E_{q^{(a)}}\ e^{iq^{(a)} \cdot y^{(a)}}\right)\prod_{a=1}^{b'}\int\left(d^3z^{(a)}\ 2E_{r^{(a)}}\ e^{ir^{(a)} \cdot z^{(a)}}\right) \\ \langle 0|\prod_{a=1}^{b'}\left(\phi[t;z^{(a)}]\right) U_{int}(t_1,t_0) \prod_{a=1}^{b}\left(\phi[t;y^{(a)}]\right)|0 \rangle $$
and the last term looks like
$$ \langle 0|T\phi_I\cdots\phi_I U_{int}(t_1,t_0)|0\rangle $$
with no division by $\langle 0|S|0 \rangle$ or even $t_{0,1} \to \pm \infty,\ U_{int}(t_1,t_0) \to S$ necessary.
My question is what is wrong with the way we constructed $P$ that requires taking the infinite limit in time and dividing out the vacuum bubbles ($\langle 0|S|0 \rangle$)? As far as I can tell none of this is necessary; we choose $|i\rangle$ and there's nothing stopping us from choosing it to be a free field eigenstate, and we choose $|f\rangle$ similarly. The only pitfalls I can see are if $|i\rangle$ is orthogonal to the entire interacting eigenbasis (which would make it incomplete as a basis) or that $P=0$ because evolution under the interacting Hamiltonian takes $|i\rangle$ far away from the non-interacting states.
