Weinberg, QFT 2, in Section 16.1 in a footnote 2 refers to Coleman, Aspects of Symmetry, p. 135-6, which features the $\hbar$/loop expansion. See also Refs. 3 & 4 for a similar idea. In this answer we provide a non-inductive argument along these lines. A nice feature of this argument is that we do not have to deal explicitly with pesky combinatorics and symmetry factors of individual Feynman diagrams. This is already hardwired into the formalism.
A) Let us first recall some basic facts from field theory. The classical (=$\hbar$-independent) action
$$S[\phi]~\equiv~\underbrace{\frac{1}{2}\phi^k (S_2)_{k\ell}\phi^{\ell}}_{\text{quadratic part}} + \underbrace{S_{\neq 2}[\phi]}_{\text{the rest}}, \tag{A1}$$
is the generating functional for bare vertices (and inverse bare propagator $(S_2)_{k\ell}$).
The partition function/path integral is
$$\begin{align} Z[J] ~:=~&\int \! {\cal D}\frac{\phi}{\sqrt{\hbar}}~\exp\left\{ \frac{i}{\hbar} \underbrace{\left(S[\phi]+J_k \phi^k\right)}_{=:~S_J[\phi]}\right\}\tag{A2} \cr
~\stackrel{\begin{array}{c}\text{Gauss.}\cr\text{int.}\end{array}}{\sim}&
{\rm Det}\left(\frac{1}{i} (S_2)_{mn}\right)^{-1/2}\cr
&\exp\left\{\frac{i}{\hbar} S_{\neq 2}\left[ \frac{\hbar}{i} \frac{\delta}{\delta J}\right] \right\}\cr
&\exp\left\{- \frac{i}{2\hbar} J_k (S_2^{-1})^{k\ell} J_{\ell} \right\}\tag{A3}\cr
~\stackrel{\begin{array}{c}\text{WKB}\cr\text{approx.}\end{array}}{\sim}&
{\rm Det}\left(\frac{1}{i}\frac{\delta^2 S[\phi[J]]}{\delta \phi^m \delta \phi^n}\right)^{-1/2}\cr
&\exp\left\{ \frac{i}{\hbar}\left(S[\phi[J]]+J_k \phi^k[J]\right)\right\}\cr
&\left(1+ {\cal O}(\hbar)\right) \tag{A4}\end{align} $$
in the stationary phase/WKB approximation $\hbar\to 0$. In eq. (A4)
$$ J_k~\approx~-\frac{\delta S[\phi]}{\delta \phi^k} \qquad \Leftrightarrow \qquad \phi^k~\approx~\phi^k[J] \tag{A5}$$
are the Euler-Lagrange (EL) equations for the quantum field $\phi^k$.
Notice in the diagram expansion (A3) how a bare vertex comes with $\hbar$-weight $=-1$; an internal bare propagator $(S_2^{-1})^{k\ell}$ comes with $\hbar$-weight $=+1$; and an external leg comes with $\hbar$-weight $=0$.
The linked cluster theorem states that the generating functional for connected diagrams is
$$ W_c[J]~=~\frac{\hbar}{i}\ln Z[J], \tag{A6}$$
cf. e.g. this Phys.SE post. Note that the connected vacuum bubbles $W_c[J\!=\!0]=\frac{\hbar}{i}\ln Z[J\!=\!0]$ by definition is correlated to the normalization of the path integral, and hence is not physically relevant. (We allow the possibility that it is non-zero to be as general as possible.)
Next recall the $\hbar$/loop-expansion
$$ L~=~I-V+1, \tag{A7} $$
cf. my Phys.SE answer here.
The $\hbar$/loop-expansion together with eqs. (A4) & (A6) imply that the generating functional
$$ W_{c}^{\rm tree}[J]~\stackrel{(A4)+(A6)}{=}~S[\phi] + J_i \phi^i \tag{A8}$$
for connected tree diagrams is the Legendre transformation of the classical action. Note that the EL eqs. (A5) are compatible with this.
Eqs. (A3) and (A6) yield
$$\begin{align} &W^{\rm tree}_c[J]~\stackrel{(A3)+(A6)}{=}\cr \lim_{\hbar\to 0} \frac{\hbar}{i}& \ln\left( \exp\left\{ \frac{i}{\hbar} S_{\neq 2}\left[ \frac{\hbar}{i} \frac{\delta}{\delta J}\right] \right\}
\exp\left\{- \frac{i}{2\hbar} J_k (S_2^{-1})^{k\ell} J_{\ell} \right\} \right). \end{align}\tag{A9}$$
Notice how eq. (A9) only refers to objects in eqs. (A1) & (A8), and hence can be viewed as a consequence of them alone.
Eq. (A9) realizes the fact that given an arbitrary finite set of external source insertions, then (a sum of all possible) connected tree diagrams is (a sum of all possible) trees of bare propagators $(S_2^{-1})^{k\ell}$ and bare vertices.
Note that the one-loop square root factors in eqs. (A3) & (A4) do not affect the zero-loop/tree formula (A9) & (A8), respectively.
$\downarrow$ Table 1: Structural similarity between Sections A & B.
$$ \begin{array}{ccc} A &\leftrightarrow & B \cr
\phi^k&\leftrightarrow & \phi_{\rm cl}^k \cr
S[\phi]&\leftrightarrow &\Gamma[\phi_{\rm cl}]\cr
\hbar&\leftrightarrow &\hbar^{\prime} \cr
Z[J]&\leftrightarrow &Z_{\Gamma}[J]\cr
W^{\rm tree}_c[J]&\leftrightarrow &W_c[J]
\end{array}$$
B) Finally let us address OP's question. Consider the effective/proper action
$$ \Gamma[\phi_{\rm cl}]~\equiv~\underbrace{\frac{1}{2}\phi_{\rm cl}^k (\Gamma_2)_{k\ell}\phi_{\rm cl}^{\ell}}_{\text{quadratic part}} + \underbrace{\Gamma_{\neq 2}[\phi_{\rm cl}]}_{\text{the rest}}.\tag{B1}$$
Unlike the classical action (A1), the effective action (B1) depends (implicitly) on Planck's reduced constant $\hbar$. We would like to make a loop-expansion wrt. a new parameter $\hbar^{\prime}$.
To this end, define a partition function/path integral
$$\begin{align} Z_{\Gamma}[J] ~:=~&\int \! {\cal D}\frac{\phi_{\rm cl}}{\sqrt{\hbar^{\prime}}}~\exp\left\{ \frac{i}{\hbar^{\prime}} \underbrace{\left(\Gamma[\phi_{\rm cl}]+J_k \phi_{\rm cl}^k\right)}_{=:~\Gamma_J[\phi_{\rm cl}]}\right\} \tag{B2}\cr
~\stackrel{\begin{array}{c}\text{Gauss.}\cr\text{int.}\end{array}}{\sim}&
{\rm Det}\left(\frac{1}{i} (\Gamma_2)_{mn}\right)^{-1/2}\cr
&\exp\left\{ \frac{i}{\hbar^{\prime}} \Gamma_{\neq 2}\left[ \frac{\hbar^{\prime}}{i} \frac{\delta}{\delta J}\right] \right\}\cr
&\exp\left\{- \frac{i}{2\hbar^{\prime}} J_k (\Gamma_2^{-1})^{k\ell} J_{\ell} \right\}\tag{B3}\cr
~\stackrel{\begin{array}{c}\text{WKB}\cr\text{approx.}\end{array}}{\sim}&
{\rm Det}\left(\frac{1}{i}\frac{\delta^2\Gamma[\phi_{\rm cl}[J]]}{\delta \phi_{\rm cl}^m \delta \phi_{\rm cl}^n}\right)^{-1/2}\cr
&\exp\left\{ \frac{i}{\hbar^{\prime}}\left(\Gamma[\phi_{\rm cl}[J]]+J_k \phi_{\rm cl}^k[J]\right)\right\}\cr
&\left(1+ {\cal O}(\hbar^{\prime})\right) \tag{B4}\end{align} $$
in the stationary phase/WKB approximation $\hbar^{\prime}\to 0$. Also the EL eqs. for the effective action $\Gamma_J[\phi_{\rm cl}]$ for the classical field $\phi_{\rm cl}^k$ read
$$ J_k~\approx~-\frac{\delta \Gamma[\phi_{\rm cl}]}{\delta \phi_{\rm cl}^k} \qquad \Leftrightarrow \qquad \phi_{\rm cl}^k~\approx~\phi_{\rm cl}^k[J]. \tag{B5}$$
Recall that the effective action (B1) is by definition the Legendre transformation of the generating functional
$$ W_{c}[J]~\equiv~\Gamma[\phi_{\rm cl}] + J_k \phi_{\rm cl}^k \tag{B8}$$
for connected diagrams. Note that the EL eqs. (B5) are compatible with this.
Due to the structural similarity between two the Legendre transformations (A8) & (B8), cf. Table 1, we obtain an analogue to eq. (A9):
$$\begin{align}&W_c[J]~\stackrel{(B3)+(B4)+(B8)}{=}\cr \lim_{\hbar^{\prime}\to 0} \frac{\hbar^{\prime}}{i}& \ln\left( \exp\left\{ \frac{i}{\hbar^{\prime}} \Gamma_{\neq 2}\left[ \frac{\hbar^{\prime}}{i} \frac{\delta}{\delta J}\right] \right\} \exp\left\{- \frac{i}{2\hbar^{\prime}} J_k (\Gamma_2^{-1})^{k\ell} J_{\ell} \right\} \right) .\end{align}\tag{B9}$$
In retrospect, eq. (B9) can be viewed as a functorial consequence of eqs. (B1) & (B8) alone.
On the other hand, given an arbitrary finite set of external source insertions, then (a sum of all possible) connected diagrams is (a sum of all possible) trees of full propagators$^{\dagger}$ $(\Gamma_2^{-1})^{k\ell}$ and (amputated) 1PI vertices, cf. Lemma 3.11 in Ref. 5.
Together with eq. (B9), we conclude that the effective action $\Gamma[\phi_{\rm cl}]$ is the generating functional for (amputated) 1PI vertices (and inverse full propagator $(\Gamma_2)_{k\ell}$). $\Box$
References:
S. Weinberg, Quantum Theory of Fields, Vol. 2, 1995; Section 16.1.
S. Coleman, Aspects of Symmetry, 1985; p. 135-6.
M. Srednicki, QFT, 2007; Chapter 21. A prepublication draft PDF file is available here.
D. Skinner, QFT in 0D, p. 32. (Hat tip: The Last Knight of Silk Road.)
P. Etingof, Geometry & QFT, MIT 2002 online lecture notes; Sections 3.11 & 3.12. (Hat tip: Abdelmalek Abdesselam.)
R. Kleiss, Pictures, Paths, Particles, Processes, Feynman Diagrams and All That and the Standard Model, lecture notes, 2013; section 1.5.2.
--
$^{\dagger}$ Fine print:
Assume that the generator $W_c[J]$ of connected diagrams has no terms linear in $J$, so that the effective action $\Gamma[\phi_{\rm cl}]$ has no terms linear in $\phi_{\rm cl}$, and so that $(\Gamma_2^{-1})^{k\ell}=-(W_{c,2})^{k\ell}$ is the full connected propagator, cf. my Phys.SE answer here.
Here the notion of the one-particle irreducible (1PI) vertices are defined wrt. to full propagators $(W_{c,2})^{k\ell}$, which is equivalent to the notion of 1PI vertices wrt. to bare propagators $(S_2^{-1})^{k\ell}$, cf. e.g. this Phys.SE post.
Best Answer
The Legendre transformation between the generating functional $W_c[J]$ for connected diagrams and the the effective/proper action $\Gamma[\phi_{\rm cl}]$ also works for non-zero vacuum expectation value $\langle \phi \rangle_{J=0}$. The first few orders are e.g. worked out in my Phys.SE answer here.