What are the compulsory tasks of a charge controller for a lead acid battery?
The bare minimum requirements I would set at:  


Undervoltage protection. Do not charge is battery open cell voltage is too low.  
Overvoltage protection. To protect the charger.  
Timeout. When during charging the voltage does not rise as expected.  


But you can do many more things:


You can follow the proper lead-acid battery charging profile.

Source
You can measure battery temperature, and adjust charging rate respectively.
You can give a battery end-of-life warning.
You can measure the energy going in/out of the battery. Giving a pre-warning: "please charge", and disconnect if it gets very low. In order to preserve the battery. Batteries last longer when not depleted often.