What are the benefits of a unified framework? You are right that we are rapidly going into some much used special cases line logistic regression or Poisson regression, but there is still benefit in having a common framework.

* Technology transfer from the *general linear model* (**not** generalized!), that is with gaussian errors and identity link. A lot of what one has learned from there can be directly used with glm's, especially all the modeling trick constructing the model matrix $X$. 

* A common estimation algorithm (IRLS, iteratively reweighted least squares, see https://stats.stackexchange.com/questions/236676/can-you-give-a-simple-intuitive-explanation-of-irls-method-to-find-the-mle-of-a/237384#237384), leads to a common implementation framework. This is maybe not a mathematical advantage, but software implementation and modeling advantage. It is also a teaching advantage!

* Many common concepts applied to all or most of the special cases, like link function, offsets, variance function, mean function, ... , quasi-likelihood 

* One area with little advantage of the common framework is residual analysis, which really need to be studied for each case separately. See for instance https://stats.stackexchange.com/questions/374452/family-of-glm-represents-the-distribution-of-the-response-variable-or-residuals/374461#374461

* The Nelder & Wederburn paper introding the glm is [here at JSTOR](https://www.jstor.org/stable/pdf/2344614.pdf) and the original motivations can be found there. 

A lot of information can be found at Cross Validated, see as a start https://stats.stackexchange.com/questions/104399/why-do-we-use-glm