Dirichlet Generalized Additive Model

Dear users,

I want to use Dirichlet regression in order to model proportions of deaths, using i.e. Age, Years, and Gender as predictors. I gave up this idea a few months ago, due to the impressive (for my modest background) computational and coding effort required. But now I have just discovered this link:

This is amazing and it works well on my data, I have also tried the GAM regression, with Dirichlet assumption, and is ok!
Given that, I would like to ask your opinion about the theoretical background behind a Dirichlet regression (GLM or GAM), even more considering the controversy if we can consider Beta and Dirichlet regression as a true GLM - your opinion on this is welcome.
To be pragmatic, how should I specify (on paper) the equations that are related to the bmrs code for Dirichlet GLM and/or GAM?

Many thanks for the outstanding work of this comunity.