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! This program is not a 100% success, as for some important glm's this algorithm do not work very well, as witnessed by the paper logbin: An R Package for Relative Risk Regression Using the Log-Binomial Model.
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 of glm represents the distribution of the response variable or residuals. But see here for an approach using simulated residuals.
The Nelder & Wederburn paper introding the glm is here at JSTOR 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