First, it's important to know what kind of states there are. I'll call them ground state $|0\rangle$, one-particle state $|k\rangle$ and multi-particle state $|n,k\rangle$. Here, $k$ specifies the momentum of a particle and $n$ is a collection of other quantum numbers that may arise (e.g. relative momentum between some other momenta, ...).
We want $a^\dagger|0\rangle$ to create a one-particle state. In Eq. (3.19), Srednicki defines
$$ \varphi(x) = \int \widetilde{\text{d}k} \big[ a(k)\, \text{e}^{\text{i} kx} + a^\dagger(k)\, \text{e}^{-\text{i} kx} \big].\tag{1}\label{a} $$
Assume, that $a^\dagger$ creates both a one-particle state, as well as the ground state, like this
$$ a^\dagger |0\rangle = c_1 |k\rangle + c_2 |0\rangle, $$
with some complex numbers $c_{1,2}$.
Now we calculate the vacuum expectation value of $\varphi$ (and only consider the $a^\dagger$-part for notational simplicity):
\begin{align*}
\lim\limits_{x\to 0}\langle 0|\varphi(x)|0\rangle &= \lim\limits_{x\to 0}\int \widetilde{\text{d}k} \langle 0| a^\dagger(k)\, \text{e}^{-\text{i} kx} |0\rangle \\
&= \int \widetilde{\text{d}k} \langle 0| a^\dagger(k) |0\rangle \,\text{e}^{0} \\
&= \int \widetilde{\text{d}k} \langle 0| \big[ c_1 |k\rangle + c_2 |0\rangle \big]\\
&\propto c_1 \underbrace{\langle 0|k\rangle}_{0,\text{ orthogonal!}} + c_2 \underbrace{\langle 0|0\rangle}_{1}\\
&= c_2 \neq 0
\end{align*}
This means, if we let $a^\dagger$ create both a one-particle state and the ground state, then $\langle 0|\varphi(0)|0\rangle$ does not vanish. But if we define $\langle 0|\varphi(0)|0\rangle\equiv 0$, we must set $c_2=0$, which means that $a^\dagger$ truly only creates a one-particle state in Eq. \eqref{a}.
Ad comment: (1) If we want to be precise and include the $a$ term, we should also not perform the limit so quickly. Inside the integration, there's a $\text{e}^{\text ikx}$ and a $\text{e}^{-\text ikx}$. They are linearly independent, so each should vanish. (Also note that while $x\to 0$, $k\to \infty$, so that's ill-defined within the exponential)
(2) $a$ being evaluated at $t=\pm\infty$ helps us make it non-interacting. If this doesn't answer your question, I am sorry I did not understand it. If so, please restate it.
