beta “limits” you to [0,1], its based upon gamma distribution.
See for understanding the beta distribution:
http://varianceexplained.org/r/beta_binomial_baseball/
How come something quite common such as softmax/dirichlet regression has never been encountered/established in stan-like languages?
softmax is implemented in Stan and used by some models in Stan. I have no time to do a
research, but I used it myself. Dirichlet regression I have never seen.
BTW, there are interpretations of exponential, gamma, weibull, lognormal distributions and connections
eg. the exponential and the poisson. Normally we define a problem and then we think about, but you came
with the tool first and then ask how you can use it, like how I can nail that screw with my hammer.
Sorry, if I sound judging. Just my 2 cents.
https://cran.revolutionanalytics.com/web/packages/betareg/vignettes/betareg.pdf