Wouldn't it be better to use poisson regression for count data? Also in logistic regression what is the advantage of using the log link versus the logit link? I know you can get the log relative risk with the log link. But why use relative risks as opposed to odds ratios?
Your question starts with a premise, namely that people actually use logistic regression for count data. I have not seen so, except when employing a hurdle model. Logistic (and probabilistic) models are designed for binary dependent variables. Because of this, the coefficients (which are odds ratios) can be transformed to marginal effects on probability of having a 1 in the dependent variable. I don't see how this can be meaningful for count data. I also think that you run into errors when you regress a logistic model with count data that actually has no 0 values - which is possible.
Also, Poisson regression is not the only possibility to deal with count data. There is negative binomial as well. The difference between these two: Poisson restricts the first two moments (mean and variance) to be equal, while negative binomial doesn't.