Does survival analysis implicitly do classification? I've seen a lot of applications of survival analysis in predicting death under certain medical conditions. Obviously, death is guaranteed eventually. For problems where the event of interest is not guaranteed (if cancer is recurring or not), is survival analysis still applicable? i.e. does it implicitly perform classification as well as predicting the survival function?
If not, would a two-stage model be more appropriate? First, training a model to classify if the cancer will recur or not and then multiplying this probability by the output of the survival model.

What you describe is typically called a cure model in survival analysis. A strict two-stage analysis like you describe isn't the best approach, as for those still event-free at late times you don't know if that's because they are in the "cured" group or if they would have the event of interest if you waited longer. A joint approach that models both possibilities together is better.
"Cure Models as a Useful Statistical Tool for Analyzing Survival" by Othus et al, Clin Cancer Res 18: 3731–3736 (2012) outlines and illustrates such models. In practice, especially in situations with overall high survival or where the "cure" probability doesn't differ much between groups, ignoring the possibility of "cure" might not make much difference; hazard ratios still can be informative.
