Like Jon Clark said, "has anyone ever made a hero intentionally relevant?" The answer is usually if not exclusively no. More often than not you end up with strange joke of a character who's like an amalgam of "top 40 hits" in pop culture. They become destined to have time and pop culture turn on them and pass them by. Relevancy is usually made by setting trends or redefining already present trends. The character creates a sort of pocket universe in pop culture that we visit and attune our thinking to that particular universe and then enjoy the ride. Batman has been doing this since his big popularity boom in the 80's. Under a truly hawkish gaze, his world makes little to no sense and has more in line with Alice in Wonderland than it does with hard nosed crime dramas.
Shows like Adventure Time, Gumball, or Steven Universe all spun out of creators just wanting to do something interesting with what they knew. Things like D&D are not culturally relevant anymore but they are the literal crux of the wildly successful show Adventure Time.
Superman's got a much better shot at relevancy if the creators at work get more creative and outgoing with the aspects (obvious and hidden) of Superman's little pocket of pop culture. Push and polish those aspects. If you want relevancy then bring in an influx of fresh eyes who have something to say about the character.
Also this doesn't equal moral complexity. I'm not as opposed to a Superman who kills as I used to be, but the act of killing doesn't make anyone more or less complex morally. One could argue that not killing makes one pretty complex morally (seems to work for Batman) or Max Landis' idea that Superman will go for the kill if your power and perceived danger makes you an "adult" in this world of relative "children". The idea that his responsibility no longer extends to you. Both pretty interesting takes on the characters morals if handled well. Hell I've even been semi-championing a complete pacifist Superman just to see it done and done well.
But ultimately I might be wasting my breath because...