Here the most common electric fence is for property protection, and is regulated legally to no more than 8J per pulse, and a 1Hz pulse repetition rate.
Most common failures I see are the pulse transformer shorting out, almost always the final row of turns shorting, as that is the highest voltage and also the highest dVdt as well. Second is the MOV units going low resistance, and as they are there to define the peak voltage, and also have to absorb all the energy of the pulse, you have to use multiple 800V units to get the energy absorption ability, a single unit would be expensive and unreliable. Lightning does cause problems, but if installed correctly, with the required number of grounding rods at the correct spacing, they will survive local near hits with no damage, though you will often find that the power supply to them has blown a fuse. you can not say the same for your local electric supply though, or your phone lines if you still have them.
Almost all of the energisers use a battery, as the first couple of cycles on the internal charger is driving a short circuit, so it does draw a heavy current, in the order of 10A for a very short time, to start charging that capacitor. Might be 1A average over the cycle, but those do not run without a battery, even though the battery charger built into them is capable of delivering 3A, the unit will trip out with low battery.
I have taken apart the old analogue only units, and the LED in the optocoupler is used with the phototransistor to charge a small capacitor on the controller, and this voltage is then used to drive the voltage display, generally a set of comparators driving LED's to show low, med and OK, with the capacitor being discharged by a small transistor before the next pulse. Modern units have the optical path as a box in the case, or as shielded plastic partitions between lid and base moulding, so there is a large gap, as that optocoupler is running at the fence voltage, and no optocoupler cheaply available will handle 10kV pulses.
Designed to be serviced, the parts that fail most commonly are either plug in, or a separate board, or are on large pads so you can easily replace them, and almost all faults are on the high voltage side, the low voltage side is quite reliable and rugged.