I have not read the paper you linked to in depth, but whether you use a multi-species vs. multiple single-species models depends in large part on objectives as well as the characteristics of the species. The basic concept of a simple multi-species occupancy model (i.e., the one implemented by the msPGOcc function in spOccupancy) is that species-level effects are estimated as random effects from a common, community-level distribution. This assumes that species-level effects across the group of species included in the model can reasonably be assumed to be normally distributed around some community-wide average. If your community of species is not likely to meet this assumption, then you may consider using single-species models if you have enough data. Multi-species occupancy models provide improved precision at the risk of potentially lower accuracy for certain species (e.g., particularly rare species or species that have vastly different covariate effects compared to the rest of the community). This sort of bias vs. precision tradeoff is a classic tradeoff in statistics. You do not need to only include species that respond similarly to covariates in a multi-species model. The large variance also does not imply that you need to rethink the species in the model. Again, the assumption of a multi-species model is that the species-level effects can be assumed to be normally distributed around some mean. If that is the case, it is perfectly feasible for there to be a large variation across species in the community. Instead of trying to follow some pre-specified rule, I would encourage you to consider the assumptions that go into the multi-species model and the implications this has on your inferences. If your primary interest is on interpreting the effects of some covariate on individual species and you expect some of these species to be vastly different from the majority of the species in the community, then you will likely be better off fitting a single-species model. If you are interested in both species-level effects, how these vary across the community of species, and perhaps also in predicting community-level patterns (e.g., richness, diversity), then multi-species models are likely the way to go.
Here is another good paper on multi-species modeling that might be relevant as you think about this.