I am trying to wrap my head around GMM estimation of a single factor model such as the CAPM. I started by asking How come the cross-sectional CAPM equation produces $N$ moment conditions (not $1$)? and am now following it up. Equation $(12.23)$ in Cochrane "Asset Pricing" (2005) section 12.2 (p. 241) says the moments are $$ g_T(b) = \begin{bmatrix} E(R^e_t-a-\beta f_t) \\ E[(R^e_t-a-\beta f_t)f_t] \\ E(R^e-\beta \lambda) \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \tag{12.23}. $$ where $R^e_t=(R^e_{1,t},\dots,R^e_{N,t})'$ is a vector of individual assets' excess returns, $\beta=(\beta_1,\dots,\beta_N)'$ is a vector of betas, $f_t$ is factor's excess return and $\lambda=E(f)$ is the expected value of the factor's excess return. The first two rows correspond to time series regressions for $N$ assets (one regression per asset), so there are actually $2N$ conditions. If I understand correctly, the third row corresponds to a cross-sectional regression of time-averaged returns: $$ E_T(R^{ei})=\beta_i' \lambda+\alpha_i, \quad i=1,2,\dots,N. \tag{12.10} $$ ($(12.10)$ is specified for potentially many factors, but $(12.23)$ considers the simple case of a single factor, so vector $\beta'$ turns into scalar $\beta$, and the same holds for $\lambda$.)
Since we are trying to estimate (among other things) the expected market excess return $\lambda$, an obvious approach would be to use the sample average of the market excess returns over the $T$ time periods. In a GMM, the corresponding moment condition would be $\frac{1}{T}\sum_{t=1}^T (f_t-\lambda)=0$. In the notation of Cochrane, it would be $\color{blue}{E_T(f_t-\lambda)=0}$. Yet this is not what we see in the GMM estimator for the single-factor model in equation $(12.23)$.
What I would find intuitive is to have $$ \tilde g_T(b) = \begin{bmatrix} E(R^e_t-a-\beta f_t) \\ E[(R^e_t-a-\beta f_t)f_t] \\ \color{blue}{E(f_t-\lambda)} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \color{blue}{0} \end{bmatrix} \tag{12.23'} $$ or perhaps $$ \tilde{\tilde g}_T(b) = \begin{bmatrix} E(R^e_t-a-\beta f_t) \\ [E(R^e_t-a-\beta f_t)f_t] \\ \color{red}{E(R^e-\beta \lambda)} \\ \color{blue}{E(f_t-\lambda)} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \color{red}{0} \\ \color{blue}{0} \end{bmatrix}. \tag{12.23''} $$ Again, since we are interested in estimating $\lambda$ (among other things), is there a reason for excluding the obvious moment condition in blue from $(12.23)$? I do not have well developed intuition around GMM, but I think the blue condition is pretty informative about $\lambda$, probably more so than the red one.