Superman is a hard character to write well. I've thought that for a long time. It's why, out of all the live action films, only two of them actually work. And even then have some issues.
TL;DR version-- The character is not the problem, the people behind him are.
I think lacking a consistent and confident vision-- while sticking to the core of the character-- is the biggest part of the problem*. Every time he's had that, he's thrived.
From '38-'86, he had that. When he changed from a low-powered social crusader to a planet throwing space adventurer, it happened organically over decades. When DC first hit the reboot switch, and brought his powers down to more manageable levels and make him more "relatable" (aka Marvel-ize him)*, Byrne still stuck to his core principals and
attitude... he just took a different route to get there.** This was carried on nicely by Jurgens, Ordway, Stern etc. It was only Loeb and Waid trying to reincorporate Silver Age trappings that first rocked the boat (even though I agree with them both in terms of what I want from the character) from a continuity perspective. From then on, I think the biggest issue has been DC listening to the detractors ("He's not kewl, he's out of date")***, putting lesser creatives on him (obvious exceptions aside) and chopping and changing constantly... In other words, instead of realising that poor management was the problem, they decided the character was.
The same applies to the movies. The problem lies not with the character, per say... People loved the first two movies because they were good movies... They didn't like the rest because they weren't good, to varying levels. Once Donner was gone, the Salkinds had Lester push the series closer to their original vision aka goofy comedy. Audiences didn't like it. The fourth movie was saddled with a poor budget and script. Same result (a death knell this time, too). The character himself was always loved or liked through all this.
SR comes around and... The only people who blame the character himself for this one are a segment of the fans, and unfortunately, WB execs (who have long since proven they don't have a clue about him). For audiences? It was boring, aiming for the wrong demographic (though I've always liked it).
In the Snyder years, things get more interesting, because the character himself is part of the core problem. MoS, in spite of having so many great ideas and good aspects... was poorly executed on too many levels, and had some bone-headed decisions. Audiences were torn/indifferent. The biggest reaction to the character himself comes from BvS. Amongst the huge problems it has with virtually all the other characters, the writing and direction in general, one of the biggest complaints is this -- "Why is Superman such a morose, stone-faced ***hole?" In other words... They wished he'd been MORE LIKE SUPERMAN.
For both comics and movies-- Bring back quality, consistency and confidence that adheres to the core character, I believe people will come. DC Rebirth (in spite of continuity still being all over the place), is a huge step in the right direction so far.
*Though I don't believe it's the core problem, I concede that writing the SUPER SUPER Superman would be a challenge for an extended run. A challenge, but not impossible with the right teams of people.
** Byrne's run-- in spite of being good quality fun that I appreciate-- was responsible for one of the more damaging changes to Superman amongst the fandom... "Clark Kent is who I am, Superman is what I can do". I believe this is fundamentally missing the point of the character, and destroying a lot of what makes him so interesting, complex and unique. It made him a little more pedestrian... a little more mundane. Thankfully though, he upheld the other aspects of him quite well.
*** Which, as almost always, leads to WB/DC over-correcting. First by making Superman himself near constantly wondering to himself "Why am here?! What am I supposed to do?! AM I A GOOD SUPERHERO?!!" from about '04 to damn near June/July this year (thank you, Rebirth)... Then when they reboot again, over-correcting to "Superman's kewl, armour wearing, and kind of a jerk" with the New 52. It wasn't like that all the way through, but nobody really knew what to do with Young Social Crusading Brawler Superman outside of Morrison and sometimes Pak. Everyone else... sucked to various degrees, and DC's marketing of him was woeful (to the point where I myself have only JUST started reading him).