I would like to say that the original proof of Serre (in Corps Locaux, V ยง7, p. 102) is really easy and elegant, so I'm going to present my understanding of it here for anyone who does not pursue Yoshida's method.
Suppose that $G_n \neq G_{n+1}$, then $G_n/G_{n+1}\neq e$. There is a subgroup $H\subset G/G_{n+1}$ such that $G_n/G_{n+1}\not\subset H$ and that $(G/G_{n+1})/H$ is cyclic ($\star$$\star$). By Herbrand's theorem we have
$$(G/G_{n+1})_{\varphi_{G_{n+1}}(n)} = G_nG_{n+1}/G_{n+1} = G_n/G_{n+1},$$
$$(G/G_{n+1})_{\varphi_{G_{n+1}}(n+\varepsilon)} = G_{n+\varepsilon}G_{n+1}/G_{n+1} = e, \forall \varepsilon>0;$$
and
$$((G/G_{n+1})/H)_{\varphi_H(\varphi_{G_{n+1}}(n))} = (G_n/G_{n+1})_{\varphi_{G_{n+1}}(n)}H/H = (G_n/G_{n+1})H/H\neq e,$$
$$((G/G_{n+1})/H)_{\varphi_H(\varphi_{G_{n+1}}(n+\varepsilon))} = (G_n/G_{n+1})_{\varphi_{G_{n+1}}(n+\varepsilon)}H/H = e, \forall \varepsilon>0.$$
By continuity of $\varphi_H\circ\varphi_{G_{n+1}}$ we have $\varphi_H(\varphi_{G_{n+1}}(n))\in\mathbb{Z}_{\ge 0}$. By the case of cyclic groups we have $\varphi_{(G/G_{n+1})/H}(\varphi_H(\varphi_{G_{n+1}}(n)))\in\mathbb{Z}_{\ge 0}$, and by Lemma 6.10,
$$\varphi_{(G/G_{n+1})/H}\circ \varphi_H\circ\varphi_{G_{n+1}} = \varphi_{G/G_{n+1}}\circ\varphi_{G_{n+1}} = \varphi_G,$$
and we are done.
($\star$$\star$) Lemma. Suppose that $G$ is a finitely-generated abelian group, $G'\neq e$ is subgroup of $G$, then there is a subgroup $H$ of $G$ such that $G'\not\subset H$ and that $G/H$ is cyclic.
Proof. By the structure theorem of finitely-generated abelian group we have $G = C_1\times \cdots \times C_r$ for cyclic $C_i$. For $1\neq g\in G'$, write $g = g_1\cdots g_r$ for $g_i\in C_i$, then $g_{i_0}\neq 1$ for some $i_0$. Pick $H = \displaystyle{\prod_{i\neq i_0}} C_i$, then $G/H\cong C_{i_0}$ is cyclic, and $g\notin H$ because $gH = g_{i_0}H\neq H$.
Best Answer
Self-answering (should have done a bit ago) as the question served its purpose quite a bit ago:
A proof is available in Class Field Theory notes by James Milne. The same proof is also explained in the Lecture 23, MIT OCW, 18.786 Number Theory II.