This is a really good question!
When it come to anomaly or black swan events, in fact many phenomena are not following a Gaussian distribution, but they have havier tails (e.g. like expected returns in the stock market or medical health indicator data).
So using fat tail distributions like student t-distribution can absolutely work for anomaly detection and often it makes much sense since especially the long tail events come with significant costs if not detected (the cost of the false negatives). Here is a paper using generalised student t approach for anomaly detection which could be interesting to take a look at: https://people.cs.vt.edu/~clu/Publication/2013/AAAI-Lu-2013.pdf
Another note on anomaly detection models, e.g. vanilla variational autoencoders rely on a Gaussian prior in the latent space in general but there are also extensions discussed in literature e.g. to fat tail distributions like student t distribution: [2004.02581] Variational auto-encoders with Student's t-prior
Hope that helps, @Terry_Green!
Best regards
Christian