The issues that Superman faces is indicative of the glaring issues that DC has continued to face for decades now. It isn't that audiences have outgrown the character or that he is "too good" for us. To over simplify matters would be to say that DC has failed as a publisher to reinvent it's overall approach to their iconic lineup of characters for the sake of modern storytelling sensibilities.
In the 70s and 80s there was an editorial effort at DC to reinvigorate their lineup to reflect what they perceived was Marvel's key to success. Batman returned to a more pulpish tone and style (even more so after Frank Miller got to work on the character). Socio-political tales were weaved into heroics in books like Green Lantern/Green Arrow. They even began a focus on more serialized tales (DC maintained their traditional one-and-done paradigm for quite some time).
Despite some successful retoolings, the overall product line was still falling short of that being done at Marvel. A big reason for that is this continued issue of refusing to really do anything different with the characters. Yes, some Elseworld tales have since seen some unique spin on the concepts, but generally writers are incredibly limited as to where they can take these iconic characters.
Notice that the vast majority of work from DC is very plot driven rather than character driven. A Batman or Superman story is more often than not all about the events occurring on the page rather than the impact it has on the characters. How often do we see the day in the life of Bruce Wayne, the psychologically and emotionally damaged man? How often do we see a story where we explore how lonely and broken a man Bruce is?
This doesn't happen by editorial design. The characters are sealed away from such explorations for fear of tarnishing their iconic imagery.
Now compare this to the work at Marvel, where many of their stories are actually about the characters themselves. We see the lives of the Fantastic Four at home, how their relationships with each other and others effect them both personally and out in the field. Peter Parker is an emotional roller coaster and his life sometimes has him making some questionable decisions. He makes mistakes. He suffers. And others suffer. They grow and change. Not always permanent, but layers and wrinkles are certainly added.
The key to great character work is to dig deep, flesh them out to the point where they aren't simply "characters" but people. Highs and lows. Strengths and weaknesses. Hopes and dreams. Selfish desires.
People will point to All-Star Superman as a high point, but all it does is celebrate the icon rather than expand upon him. It is a time capsule, a love letter to the icon we all know and love...and ultimately reject for being so two-dimensional.
There are many sides to a person, and the best character work reflects this. With Marvel superheroes, there are often 3 major faces: the colorful costumed hero, the public identity, and the real man beneath it all. Spider-Man is witty and (mostly) beloved by the public. Peter Parker is often seen as a slacker, lazy and unreliable. But the real Peter is underneath all that. Both are masks for that scared, shy kid from Forrest Hills who still misses his Uncle Ben.
DC characters lack this third side, or at least it is rarely if ever allowed to be shown.
For Batman there is: the cold, calculating and obsessed detective, the affluent yet entirely fictional playboy persona, and the scarred, lonely young boy who never truly got over the death of his parents. But we never really see that tortured boy, only the cool and iconic badass who always wins.
And then there is Superman and "mild mannered" Clark Kent. The god and his goofy cover story. But there should be Clark, the small town kid from Kansas with his own hopes and dreams and fears. This Clark exists (or should exist) beneath that outter heroic smile. Beneath that slouch and glasses and clumsy stumble. This is the man we need to identify with, a man who struggles to be what the world wants and needs while somehow allowing himself to be a tad bit selfish like any normal person. Remember, he isn't a god. He's a man who happens to have godlike abilities. Even the Krytponians had their faults. They had needs and wants, selfish or otherwise. But even while he was born Kal-El, he was raised his entire life as a boy named Clark. He went to local schools. Had crushes. Felt loss and jealousy. He isn't perfect, he just does his best to live up to an ideal.
This is why Captain America has remained rather relevant all these years, even more so in the last decade or so. He was a golden age character just like Superman. A living icon of truth, justice, and the American Way. But he's also been allowed to be Steve Rogers the man, faults and all. He's made mistakes. He's been selfish. He's been more man than myth.
And that's the problem with Superman. He can crush a piece of coal into a diamond. He can fly around the Earth so fast he can turn back time. He can shoot lasers and fly and even go supernova all he wants. He can be the myth and icon for truth and justice. That's what makes him Superman and why he has continued to endure when so many others faded into obscurity. But to make him relevant requires allowing writers to share with readers the man behind the legend. The man inside all that. Show us that despite having all the powers in creation, despite fighting off even death itself, there is still that flustered farm boy afraid that he might not save the world the next time. A man who wishes to live a quiet life with a wife and family, in a world that doesn't need Superman.