Let $G$ denote the group, and suppose one has an enumeration of its irreducible representations $V_{\lambda}$ by some combinatorial objects $\lambda$, like integers for $SU_2$ (or finite dimensional non-unitary representations of $SL_2$) or integer partitions, etc. There are two different Clebsch-Gordan (CG) problems which the OP seems to conflate.
CG1) The numerical CG problem:
It is to figure out the multiplicities $m(\lambda,\mu;\nu)\in\mathbb{N}$ in the general decomposition into irreducibles of tensor products of two irreducibles:
$$
V_{\lambda}\otimes V_{\mu}=\bigoplus_{\nu}V_{\nu}^{\oplus m(\lambda,\mu;\nu)}\ .
$$
CG2) The explicit CG problem: It is to realize the above decomposition with explicit intertwiners, namely, to write a decomposition of the identity operator $I_{V_{\lambda}\otimes V_{\mu}}$ on $V_{\lambda}\otimes V_{\mu}$ in the form:
$$
I_{V_{\lambda}\otimes V_{\mu}}=\sum_{\nu}\sum_{j=1}^{m(\lambda,\mu;\nu)}
\iota_{\lambda,\mu,\nu,j}\circ\pi_{\lambda,\mu,\nu,j}
$$
where $\pi_{\lambda,\mu,\nu,j}\in {\rm Hom}_G(V_{\lambda}\otimes V_{\mu},V_{\nu})$ and $\iota_{\lambda,\mu,\nu,j}\in {\rm Hom}_G(V_{\nu},V_{\lambda}\otimes V_{\mu})$ are explicit $G$-equivariant maps, i.e., intertwiners.
Note that to be able to even ask the question, a prerequisite is to solve
CG0) The parametrization of irreducibles: Namely, understanding the list of irreducibles, and having a parametrization $\lambda\mapsto V_{\lambda}$.
For $SU_2$, $SL_2$ all these problems were solved by Paul Gordan and Alfred Clebsch in the mid 19-th century,
see Section 2 of my article:
Problem CG1 for $SU_n$, $SL_n$, $GL_n$ has been solved, and the multiplicities are the so-called Littlewood-Richardson coefficients.
For $S_n$, CG1 is much more difficult. The multiplicities are the Kronecker coefficients, and there is no satisfactory combinatorial description for them.
The recent article by Böhning and Graf von Bothmer does not just solve CG1 for $SU_3$, $SL_3$ (that's known from a long time ago), but rather problem CG2 for these groups. The case of $SL_n$, $GL_n$ is still open. When $\mu$ is the fundamental representation (adding a single box), there are some results, see
- M. Hunziker, J. A. Miller, and M. Sepanski. Explicit Pieri Inclusions. Electronic J. of Combinatorics 28 (2021), no. 3, P3.49.
and references therein (in particular some older work by Peter Olver).
As for CG0 in the case or $SL_n$, it was solved by Alfred Clebsch in the 1870's, and later by Deruyts, and then Schur, see:
For $S_n$, CG0 was solved by Alfred Young and later by Specht. A good account is
in the lectures by Adriano Grasia Alfred Young’s construction
of the irreducible representations of $S_n$.
Finally, the only instance of CG2 related work for $S_n$ that I am aware of is:
