In my answer to this question I first noted that the intersection point of three tangents to the original curve $y=x^3$ would be $(x_0,y_0)=(\frac23 (a+b+c),-2 abc).$ And then that the condition for three tangents meeting in a point $(x_0,y_0)$ would be $ab+ac+bc=0, a + b \neq 0.$ This reminded me of the elementary symmetric polynomials:
$$\pi(\zeta-a)(\zeta-b)(\zeta-c)=\zeta^3-(a+b+c) \zeta^2+ (ab+ac+bc)\zeta-abc=\zeta^3-\frac32 x_0 \zeta^2+0\zeta+\frac12 y_0.$$
Noting that the discriminant is $-27y_0(y_0-x_0^3)$ solved the problem there.
But might there be a representation theoretic way of giving meaning to this?
I know that $(t,t^3)$ is in the affine piece $x_0=1$ of the projection from $(x_0:x_1:x_2:x_3)=(s^3:s^2t:st^2:t^3), s=1$ by skipping $x_2.$ Could the representation theory of $\frak{sl}$ $V$ where $V$ is of dimension two on $s, t$ figure here? Interacting with the $\Sigma_3$-action?
Question: "Could the representation theory of sl V where V is of dimension two on s,t figure here? Interacting with the Σ3-action?"
Answer: If $V:=k\{e_0,e_1\}$ and if $V^*:=\{x_0,x_1\}$, it follows the $3$-uple embedding $v_3:C:=\mathbb{P}^1 \rightarrow \mathbb{P}^3$ is induced by the linear system $\mathcal{O}(3)$ on $C$. And $H^0(\mathcal{O}(3)) \cong Sym^3(V^*)$ is an irreducible $SL(V)$-module. You may construct "discriminantal varieties" corresponding to the embedding $v_3$ and you get a sequence of varieties $D_i(\mathcal{O}(3))$ containing $C$ - these are all related to the representation theory of $SL(V)$.
The first discriminant $D_1(\mathcal{O}(3))$ is defined as follows: Let $I \subseteq C \times \mathbb{P}(H^0(\mathcal{O}(3))^*)$ be the following incidence variety (let us speak the classical language in CHI in Hartshorne): $I$ is the set of pairs $(x, s)$ with $x\in C$ and $s\in H^0(\mathcal{O}(3))$ is a section where "locally" $s(x)=0$ and $Ts(x)=0$. Hence $s$ is a global section with a zero at $x$ and where the tangent $Ts$ is zero at $x$. We define $D_1(\mathcal{O}(3)):=p(I)$ where $p$ is the projection map. You may do something similar for "higher order tangencies" and what you get are discriminants $D_i(\mathcal{O}(3))$ for $i \geq 2$.
There is a canonical map
$$\pi^*: (\mathbb{P}^1)^{\times 3} \rightarrow \mathbb{P}^3 \cong Sym^3(\mathbb{P}^1)$$
and the inverse image(scheme) of the discriminant(s) is the diagonal. Your map $\pi$ is the map $\pi^*$ restricted to affine open subset of $\mathbb{P}^3$.
The discriminant variety $D_1(\mathcal{O}(3)$ parametrize the set of sections $s\in H^0(\mathcal{O}(3))$ with one "repeated root", the variety $D_2(\mathcal{O}(2))$ parametrize sections $s$ with one root of multiplicity $3$. Hence $D_2(\mathcal{O}(3)) \cong v_3(C)$.
You may in fact construct $C$ as a quotient $SL(V)/P$ where $P$ is a parabolic subgroup and there is a left action of $SL(V)$ on $C$ and $C^{\times 3}$. This action commutes with the action of $S_3$ - the symmetric group on $3$ elements.