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PCA or LDA for dichotomous variables?

Suppose I want to build a clustering model based on $k$ dichotomous variables. Since we observe two levels in each variable ($0$ and $1$), PCA would suffice – from here, the distances between the observations will be their dissimilarity scores, which are useful to calculate clusterings.
For LDA, we model the distribution of each variable as a mixture of two multivariate normal distributions (e.g., distribution of variables $X_1$ and $X_2$ will be $p(\vec{x_1}) = \frac{1}{2} N(\vec{0}, I_2) + \frac{1}{2} N(\vec{\mu_1}, \Sigma_1)$, with mean vector $\vec{\mu_1}$ and covariance matrix $\Sigma_1$), and then model the mean vectors as a linear combination of latent variables $\vec{l_1},…, \vec{l_k}$ (in my specific case, $\vec{l_1} = \vec{\mu_0}$, $\vec{l_2} = -\vec{\mu_1}$,…, $\vec{l_k} = -\vec{\mu_k}$).
So my question is: what’s the rationale to use LDA in a scenario like this? I’m not aware of any reason for choosing one over the other, except that we know that PCA is not
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