Single-species SDMs are common & have lots of good options/tools for different scenarios
In practice, we'll always miss important environmental drivers: Communities aren't just bags of climatically-compatible species
But when we "stack" SDMs to make predictions aobut community-level stuff, we implicitly assume no missing predictors.
Structured residuals:
When you're wrong about species 1, you're probably also wrong about all the species that correlate with it. This can cause cascading failures in predicting community-level properties, such as species richness.
There's important biology in these residuals
Leading eigenvectors of the covariance matrix represent systematic gradients in species composition, e.g. due to non-climate factors that favor one guild over another
Entries of $-\Sigma^{-1}$, i.e. the partial correlations among species, approximately describe the sign and magnitude of pairwise species interactions.
Modeling approaches that address this:
Most JSDMs assume linearity
Mistnet is nonlinear but is a huge pain in the ass