Directly assessing covariate effects through Gamma?

[MatPiq]

Hi,
I hope this question finds you well!
I'm trying to understand the intuition behind running a post-hoc regression on the posterior samples of theta. I understand that this propagates the uncertainty of theta to the regression model. What I'm trying to understand is why we cannot assess covariate effects by sampling the posterior of Gamma directly? Is it because this would not actually account for the variation of theta in the data (e.g. for a single binary covariate, observations with value 1 can have different thetas) or is it because the softmax transform makes interpretations of Gamma difficult?

Thank you!

Best, Matías