If I have to point at ONE thing which is the fundamental problem with the Oilers, it's clearly Daryl Katz. Now, I know an owner just signs cheques and doesn't really need to do anything in terms of the day-to-day, but with my E.Asian influenced mentality (and my considerable experience working at private schools and private universities), I've learned that in all organizations the "culture" and the ultimate success/failure is top-down directed. I mean, here in Japan, if there were such a high profile sports-team (say, the Tokyo Giants) who had failed so badly for so long despite high draft picks, etc., the owner would at some point shave his head and bow to fans in a formal apology wherein he basically announced that he is abysmal failure.
Owner aside, however, the fundamental problem with the Oilers' day-to-day management is drafting & development. The franchise's drafting has been horrid since about 1982. Once again, that's since 1982 -- two years before their first Stanley Cup. In more recent years, when they have drafted competently, for every one relative success (RNH, Draisaitl), there have been two developmental failures. (Chia trading away a future League MVP for a stay-at-home D-man didn't help.)
And aside from that, just looking at the current roster and team: the current biggest problem in 2019 is obviously lack of good players.