Finding any proof of succession amid radioactive ruins and scorched earth would be pretty much impossible, even with a decade to do it. Most things were destroyed during the Great War. Records like the ones you'd need need would be pretty much gone. It's more likely these people crowned a king with very limited "proof" of his lineage, and that it was an act of necessity more than anything. With the unlikelihood of finding any of the information they'd need (since said proof would be destroyed along with most of everything not in a vault), using up their limited supply of fuel to search for it would be a massive waste, to say the least. If you want this to happen mostly out of necessity for order and someone to guide them in the early days, as you said, then it's more likely they'd just pick the very first person that looked fit and had as much as a rumor backing their claim. They wouldn't spend resources on something they may not find if their intention really is to get someone up and in charge for the sake of guidance.