Could fitting a Poisson distribution be better than just counting random events? When measuring radioactive decays, the easiest way to get their frequency is to just count the number of events and divide by the time recorded. Would instead fitting the distribution of the times between events to a Poisson distribution have any benefits? Maybe extending the linearity to higher frequencies if the measuring device has a dead time where doublets can occur? Has this method been applied to any other problems?
 A: This has been done in this experiment 

Poisson statistics were studied using the radioactive decay of1 7Cs as a source. A scintillation counter measured gamma rays emitted by 137Cs as well as background from cosmic rays and other experiments in the laboratory. Data from approximate mean rates of 1, 4, 10 and 100 counts/sec was compared to both theoretical Poisson distributions and Monte Carlo simulations. The reduced chi squared values for the fits to theoretical distributions ranged from 0.57 to 2.13. The values from the Monte Carlo simulations were within the margin of statistical error of the data. Thus, it was found that gamma rays from the radioactive decay of Cs with added background events is accurately modeled by Poisson statistics, so each decay event in a bulk source is a random, independent event.

Italics mine. 
The decay rate curves for radioactive elements are what is useful for experiments,it gives how much the sample is depleted.
A: Suppose the dataset is a realisation of a random variable $X$, which is distributed according to the probability density distribution $f(X)$. In this case a fit of the distribution utilises most of the information contained in the dataset. Thus, the fit is a very efficient and reliable method to obtain "high quality" estimators -- "high quality" means, that the estimators possess "small" uncertainties. However, if you do not know for sure that the dataset stems from the distribution $f(X)$, you are going to fit the wrong distribution. In this case you obtain estimators for the wrong model. Thus, diagnostic methods of the fit are important.
The aforementioned idea is valid for all distributions. It's an integral part of probability theory as well as stated in several ISO norms for uncertainties in measurements. Thus, it is also valid for the Poissonian distribution.
The fit not only yields high quality estimators for the population parameters, it also allows to account for more complicated models. E.g. in your example, you might have background noise, decays of multiple radioactive elements, the dead time of the measurement device etc. All this could be incorporated into the fit. However, if you like to go crazy with your fit model, be aware of the bias variance trade-off: More complicated fit models yields smaller residuals, but might yield larger biases. Thus, if you choose a fancy fit model make sure you validate the model.
