Going back to FO and even to its progenitor, Wasteland, PA always had fusion power. Why fusion? Because it needed to be a power source that never ran out. If PA ever "ran out of gas", it would become just so much inert metal until such time as more fuel was acquired. That would mean recharging stations, battery life expectancy, etc. = Maintenance. Most players do NOT want to do Maintenance; they would prefer to just use something than to "waste" time taking care of it. [One of the things the devs all along -- and even now -- have NOT taken into account is the probable explosion that would occur if the "fusion bottle" gets breached. They allude to that happening with fusion-powered vehicles that get caught in a crossfire, but never if it is PA that had it's fusion bottle cracked.]
Maintenance was introduced with the Repair system in FO3 for weapons and armor. Just skimming these forums, you get the impression that the majority of players were quite glad that the Repair system was removed. EXCEPT for repairing PA pieces + the new addition of fusion cores to power PA. Which, quite honestly, requiring frequent swapouts of fusion cores is kind of ridiculous. Fusion is a process, NOT any kind of battery. It's fission-on-steroids. The appeal of fusion is that once started, it is almost entirely self-sustaining. What is needed in the way of finite power is the "ignition switch". We don't have fusion as a power source in RL because we haven't figured out how energize the internal plasma flux with a REALLY BIG jolt of energy. (And then once started, keeping that much energy safely contained in the fusion bottle.)
As for lore for fusion cores, there is none. It only appeared just now in FO4. And has been suggested, it keeps PA from being too powerful, and gives players cause to crawl out of their shell from time to time.