This answer provides an elementary version of the proof that there is no $[\![3,1,2]\!]$ code. The proof is based on the arguments in Quantum shadow enumerators and Quantum MacWilliams Identities, but uses only basic linear algebra and quantum error correction conditions (see e.g. section $7.3.3$ in these notes).
Quantum error correction conditions
Let $\mathcal{E}=\{P_0\otimes P_1\otimes P_2\,|\,P_0,P_1,P_2\in\{I,X,Y,Z\}\}$ denote the set of Pauli errors on three qubits. For $E\in\mathcal{E}$, define $\mathrm{wt}_x(E)$ as the number of qubits on which $E$ acts as $X$ and similarly for $\mathrm{wt}_y(E)$ and $\mathrm{wt}_z(E)$. Define $E$'s weight by $\mathrm{wt}(E)=\mathrm{wt}_x(E)+\mathrm{wt}_y(E)+\mathrm{wt}_z(E)$. Then $\mathcal{E}=\mathcal{E}_0\cup\mathcal{E}_1\cup\mathcal{E}_2\cup\mathcal{E}_3$ where $\mathcal{E}_k$ is the subset of $\mathcal{E}$ consisting or errors of weight $k$.
Suppose $\mathcal{C}=\mathrm{span}(|0_L\rangle, |1_L\rangle)\subset\mathbb{C}^{2^3}$ is a two-dimensional subspace of the Hilbert space of three qubits. For $E\in\mathcal{E}$, define the $2\times 2$ matrix $C_E$ by $(C_E)_{ij}=\langle i_L|E|j_L\rangle$ where $i,j=0,1$. The quantum error correction conditions (see e.g. equation $(7.36)$ and surrounding discussion in the above cited lecture notes) state that $\mathcal{C}$ is a $[\![3,1,2]\!]$ code if and only if for every single-qubit Pauli error $E\in\mathcal{E}_1$ we have $C_E=c_EI$ for some $c_E\in\mathbb{R}$. Equivalently, $\mathcal{C}$ is a $[\![3,1,2]\!]$ code if and only if for every $E\in\mathcal{E}_1$
$$
\left\|C_E-\frac12 I\,\mathrm{tr}(C_E)\right\|^2_F=0\tag1
$$
where $\|A\|_F^2=\sum_{ij}|a_{ij}|^2$ denotes the Frobenius norm of matrix $A$. We will show that there is no such code by proving a positive lower bound
$$
\sum_{E\in\mathcal{E}_1}\left\|C_E-\frac12 I\,\mathrm{tr}(C_E)\right\|^2_F > 0.\tag2
$$
Weights
Let $P$ denote the projector onto the code subspace $\mathcal{C}$ and define
$$
\begin{align}
A_k &= \sum_{E\in\mathcal{E}_k}\mathrm{tr}(EP)^2\tag3 \\
B_k &= \sum_{E\in\mathcal{E}_k}\mathrm{tr}(EPEP).\tag4
\end{align}
$$
Since $\mathcal{C}$ is two-dimensional, we see immediately that $A_0=4$ and $B_0=2$.
Quantum error correction conditions in terms of weights
Substituting $P=|0_L\rangle\langle 0_L|+|1_L\rangle\langle 1_L|$ into the formulas for $A_1$ and $B_1$, we have
$$
A_1 = \sum_{E\in\mathcal{E}_1}\left(\sum_i\langle i_L|E|i_L\rangle\right)^2
= \sum_{E\in\mathcal{E}_1}\mathrm{tr}^2(C_E)= \sum_{E\in\mathcal{E}_1}|\mathrm{tr}(C_E)|^2\tag5
$$
and
$$
B_1 = \sum_{E\in\mathcal{E}_1}\sum_{ij}|\langle i_L|E|j_L\rangle|^2 = \sum_{E\in\mathcal{E}_1}\|C_E\|_F^2.\tag6
$$
For any $2\times 2$ matrix $M=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, we have
$$
\begin{align}
\left\|M-\frac12 I\,\mathrm{tr}(M)\right\|^2_F &= \left|a-\frac{a+d}{2}\right|^2 + |b|^2 + |c|^2 + \left|c-\frac{a+d}{2}\right|^2\\
&= \frac12|a-d|^2 + |b|^2 + |c|^2\\
&= |a|^2+|b|^2 + |c|^2 + |d|^2 - \frac12|a+d|^2\\
&= \|M\|_F^2 - \frac12|\mathrm{tr}(M)|^2
\end{align}\tag7
$$
where we used parallelogram law $|x-y|^2+|x+y|^2=2|x|^2+2|y|^2$. Setting $M:=C_E$, using $(5)$ and $(6)$ and summing over $\mathcal{E}_1$, we obatin
$$
\sum_{E\in\mathcal{E}_1}\left\|C_E-\frac12 I\,\mathrm{tr}(C_E)\right\|^2_F = B_1 - \frac12 A_1.\tag8
$$
Thus, $\mathcal{C}$ is a distance-$2$ code if and only if $B_1-\frac12 A_1=0$.
$B_0$ in terms of $A_k$: basis expansion
It will be helpful to express $B_0$ and $B_1$ in terms of $A_k$. Pauli strings form an orthogonal basis, so
$$
P=\frac18\sum_{E\in\mathcal{E}}\mathrm{tr}(EP)E.\tag9
$$
Multiplying both sides by $P$ and taking the trace, we have
$$
\mathrm{tr}(P)=\frac18\sum_{E\in\mathcal{E}}\mathrm{tr}^2(EP)\tag{10}
$$
where we recognize $B_0$ on the left hand side and one eighth of $A_0+A_1+A_2+A_3$ on the right
$$
8B_0=A_0+A_1+A_2+A_3.\tag{11}
$$
However, $8B_0=16=4A_0$, so we can rewrite $(11)$ as
$$
0=-3A_0+A_1+A_2+A_3.\tag{12}
$$
$B_1$ in terms of $A_k$: counting anticommuting Pauli strings
In order to express $B_1$ in terms of $A_k$, we substitute $(9)$ into the definition of $B_1$
$$
\begin{align}
B_1&=\sum_{E\in\mathcal{E}_1}\mathrm{tr}(EPEP)\\
&= \frac{1}{64}\sum_{F,G\in\mathcal{E}}\sum_{E\in\mathcal{E}_1}\mathrm{tr}(FP)\mathrm{tr}(GP)\mathrm{tr}(EFEG)\\
&= \frac{1}{64}\sum_{F\in\mathcal{E}}\sum_{E\in\mathcal{E}_1}\mathrm{tr}^2(FP)\mathrm{tr}(EFEF)\\
&= \frac{1}{64}\sum_{k=0}^3
\left(\sum_{F\in\mathcal{E}_k}\mathrm{tr}^2(FP)\sum_{E\in\mathcal{E}_1}\mathrm{tr}(EFEF)\right).\tag{13}
\end{align}
$$
Now, we note that $\mathrm{tr}(EFEF)=\pm 8$ depending on whether $E$ and $F$ commute or anticommute. In particular, if $F\in\mathcal{E}_0$ then $\sum_{E\in\mathcal{E}_1}\mathrm{tr}(EFEF)=9\cdot 8$ since all nine elements of $\mathcal{E}_1$ commute with identity. Similarly, if $F\in\mathcal{E}_1$ then seven elements of $\mathcal{E}_1$ commute with $F$ and two anticommute, so $\sum_{E\in\mathcal{E}_1}\mathrm{tr}(EFEF)=5\cdot 8$. Continuing, we have
$$
\begin{align}
F\in\mathcal{E}_0\implies\sum_{E\in\mathcal{E}_1}\mathrm{tr}(EFEF)&=(9 - 0)\cdot 8 = 9 \cdot 8\\
F\in\mathcal{E}_1\implies\sum_{E\in\mathcal{E}_1}\mathrm{tr}(EFEF)&=(7 - 2)\cdot 8 = 5 \cdot 8\\
F\in\mathcal{E}_2\implies\sum_{E\in\mathcal{E}_1}\mathrm{tr}(EFEF)&=(5 - 4)\cdot 8 = 1 \cdot 8\\
F\in\mathcal{E}_3\implies\sum_{E\in\mathcal{E}_1}\mathrm{tr}(EFEF)&=(3 - 6)\cdot 8 = -3 \cdot 8.\tag{14}
\end{align}
$$
Substituting these results into $(13)$ and recognizing that $\sum_{F\in\mathcal{E}_k}\mathrm{tr}^2(FP)=A_k$, we obtain
$$
8B_1 = 9A_0 + 5A_1 + A_2 - 3A_3.\tag{15}
$$
Shadow
We need one more ingredient before we can prove $(2)$. Define
$$
S_0=\mathrm{tr}(PY^{\otimes 3}\overline{P}Y^{\otimes 3})\tag{16}
$$
and note that $S_0$ is the Hilbert-Schmidt inner product of two positive operators $P$ and $Y^{\otimes 3}\overline{P}Y^{\otimes 3}$. Therefore, $S_0\ge 0$. Moreover, by expanding $Y^{\otimes 3}\overline{P}Y^{\otimes 3}$ in the operator basis $\mathcal{E}$, we have
$$
\begin{align}
Y^{\otimes 3}\overline{P}Y^{\otimes 3}&=\frac18\sum_{E\in\mathcal{E}}\mathrm{tr}(EY^{\otimes 3}\overline{P}Y^{\otimes 3})E\\
&=\frac18\sum_{E\in\mathcal{E}}(-1)^{\mathrm{wt}_x(E)+\mathrm{wt}_z(E)}\mathrm{tr}(Y^{\otimes 3}E\overline{P}Y^{\otimes 3})E\\
&=\frac18\sum_{E\in\mathcal{E}}(-1)^{\mathrm{wt}_x(E)+\mathrm{wt}_z(E)}\mathrm{tr}(E\overline{P})E\\
&=\frac18\sum_{E\in\mathcal{E}}(-1)^{\mathrm{wt}_x(E)+\mathrm{wt}_z(E)}\mathrm{tr}(\overline{E}P)E\\
&=\frac18\sum_{E\in\mathcal{E}}(-1)^{\mathrm{wt}_x(E)+\mathrm{wt}_y(E)+\mathrm{wt}_z(E)}\mathrm{tr}(EP)E\\
&=\frac18\sum_{E\in\mathcal{E}}(-1)^{\mathrm{wt}(E)}\mathrm{tr}(EP)E\\
\end{align}\tag{17}
$$
which after multiplication by $P$, taking the trace and substituting from the definition of $A_k$ becomes
$$
8S_0=A_0-A_1+A_2-A_3.\tag{18}
$$
Positive lower bound
To derive the promised positive lower bound, begin by subtracting $4A_1$ from both sides of $(15)$
$$
8B_1-4A_1 = 9A_0 + A_1 + A_2 - 3A_3\tag{19}
$$
and add $(12)$ to obtain
$$
\begin{align}
8B_1-4A_1 &= 6A_0 + 2A_1 + 2A_2 - 2A_3\\
4B_1-2A_1 &= 3A_0 + A_1 + A_2 - A_3.
\end{align}\tag{20}
$$
However, $\mathrm{tr}(EP)$ is real, so $A_1\ge 0$ and using $(18)$ we get
$$
4B_1-2A_1 \ge 3A_0 - A_1 + A_2 - A_3 = 2A_0 + 8S_0\tag{21}.
$$
But $S_0\ge 0$ and we have
$$
\begin{align}
4B_1-2A_1 \ge 2A_0=8.\tag{22}
\end{align}
$$
Finally, using $(8)$, we obtain
$$
\sum_{E\in\mathcal{E}_1}\left\|C_E-\frac12 I\,\mathrm{tr}(C_E)\right\|^2_F = B_1-\frac12 A_1 \ge 2 > 0\tag{23}
$$
which demonstrates that quantum error correction conditions for a code of distance two cannot be satisfied by a two-dimensional subspace $\mathcal{C}$ of the three-qubit Hilbert space $\mathbb{C}^{2^3}$. Therefore, no $[\![3,1,2]\!]$ code exists.$\square$