Current/power requirements for electric fence?
After having watched this teardown of an electric fence controller and listening to this podcast which is an interview with a designer of electric fence controllers, I think I might have an answer.

The reasons why electric fence controllers mostly use 12 V lead-acid batteries as their power source are:


low price of these batteries
easy availability
can be charged in/from a car
can be charged using a cheap charger (of a type many farmers will already have).


Li-Ion cells are nowhere near as robust as lead-acid batteries. They require much more delicate charging and handling. A charge controller/balancer is a must-have. A battery protection circuit is also a must-have.

I also think that the electric fence controller manufacturers are quite conservative, if there's no pressing reason to change then they prefer not to. If you watch the video and listen to the podcast I linked above you'll notice how safety and reliability is very important.