Count process replaced the previous way of doing survival analysis:
In the previous section, we discussed the construction of classical like- lihoods for censored and truncated data. These likelihoods can be used to develop some of the methods described in the remainder of this book. An alternative approach to developing inference procedures for censored and truncated data is by using counting process methodology. This approach was first developed by Aalen (1975) who combined ele- ments of stochastic integration, continuous time martingale theory and counting process theory into a methodology which quite easily allows for development of inference techniques for survival quantities based on censored and truncated data.
So I'm not sure why it isn't so far fetch that people found another "better" way to replace it.
Source: SURVIVAL ANALYSIS Techniques for Censored and Truncated Data Second Edition by John P. Klein
I've only heard of empirical because I met someone that was working on it for his PhD at UCLA.
A quick google came up with this which give a reasonable explanation: