Mathematical connection between hazard functions for e.g. Cox models and expected values in GLMs I intuitively understand why the hazard function is interesting to link to patient attributes, as explained in this answer. However, is there a mathematical reason this is done? I'm particularly familiar with generalized linear models, where the expected value is modeled as being link linear. Is there a connection between the expected value in exponential dispersion families and the hazard function? perhaps through the log partition function somehow?

There are a few relevant factors historically.
At the population level, the progress from describing disease in counts to proportions to rates was based on non-mathematical grounds, but the relationship to the Poisson process does make inference easier. And for the Poisson process the hazard is the same as the rate and related simply to a generalised linear mean model.
The other key motivation is the Cox model, and it's mathematical but in a kind of sociological way.  The Cox partial likelihood estimator lets you have an infinite-dimensional parameter in the model but estimate the hazard ratios without estimating the infinite-dimensional parameter. What's more, there was almost no loss of efficiency relative to using a parametric baseline hazard. In addition to this being popular because it looked (a bit misleadingly) like a way of making weaker assumptions in data analysis, it was popular because it was mathematically  interesting. People wanted to understand it and to do the same thing for other semiparametric models. We now do understand it, but there are still few or no other examples of efficient estimators in semiparametric models that don't require estimating the infinite-dimensional parameter.
So, mathematically it's not so much the hazard that's interesting as the partial likelihood estimator.
