= Solution
The identity $X_i=\Omega_i-\Omega_{i-1}$ from part (a) shows that each <Young–Jucys–Murphy element> belongs to the <Gelfand–Tsetlin algebra>. Conversely, $Z_r$ lies in the centralizer of $\mathbb C S_{r-1}$ in $\mathbb C S_r$. Applying the preceding <Olshanskii centralizer lemma> at level $r$ gives
$$
Z_r\subseteq\operatorname{alg}(Z_{r-1},X_r).
$$
Induction, beginning with $Z_1=\mathbb C1$, proves $Z_r\subseteq\operatorname{alg}(X_1,\ldots,X_r)$. Thus
$$
\boxed{GZ_n=\operatorname{alg}(X_1,\ldots,X_n)}.
$$
Here every generated algebra is unital; $X_1=0$ causes no problem.
To deduce simple branching, use <Maschke's theorem>: complex representations of finite groups are <semisimple modules>. The <Artin–Wedderburn theorem> identifies
$$
\mathbb C S_r\cong\bigoplus_L\operatorname{End}_{\mathbb C}(L),
$$
where $L$ runs over the irreducible $S_r$-modules. Write their restrictions as
$$
\operatorname{Res}_{S_{r-1}}^{S_r}L
\cong\bigoplus_M M\otimes\mathbb C^{m_{L,M}}.
$$
By <Schur lemma>, the centralizer in the displayed algebra is
$$
Z_{(r-1,1)}\cong
\bigoplus_{L,M}\operatorname{End}_{\mathbb C}(\mathbb C^{m_{L,M}}),
$$
with zero multiplicities omitted. Part (i) makes this algebra commutative, whereas a full <matrix algebra> of size at least two is noncommutative. Consequently \b[every restriction multiplicity is $0$ or $1$]. This proves <multiplicity-free restriction> at each step, with no use of the branching rule in the centralizer proof.
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