Take those three scenes and fix them to still have impact, perhaps even more impact, without undercutting the character and the genre.
Superman stood by and let his father die. Clark would never let his father die and he's Superman, so he can save anyone from anything as long as he's close enough and knows it's going on. Clear example of the character being ridiculously overpowered and therefore hard to write well. But, this is also exactly why Pa Kent is (almost) always shown as having a heart attack and dying of natural causes. If there's some threat, Supes can always just save him. Boring. So Pa dies of natural causes. Nothing Supes can do about the normal human lifespan. This is a clear example of Chesterton's Fence. The writers wanted to make it their own and failed to understand why it was done this way for so long. Hell, you could even have Pa Kent raise his hand to say "no" only for Clark to do it anyway, then have Pa drop dead from the heart attack. Barely have to change the scene. And you don't undercut the core of the character.
Superman did basically nothing to save the civilians during the fight with Zod. Yep, this is a classic learning moment for a young Superman. But what's learned? "People die if I don't act." Okay. Is there a better way to convey that without undermining the character? You betcha. Maybe hundreds of 'em. Here's a good one: the old standby of the try-fail cycle. You have Superman, the ur-superhero...gasp...actually trying to save people. Wild concept, I know. But...you have him fail a few times. Try-fail. Emphasis on the try part. Which sets up a classic example of the villain using Superman's caring for the civilians to torture and taunt him. Zod endangers the civvies, Supes try-fail cycles through a few buildings worth of civilians, finally saves some of them in a panic, Zod notices so targets the civvies even more, Supes in now frantically on the hook and desperate to save people. Like he should...because he's a damned superhero. You have him learn the exact same lesson without undermining the character's core.
Superman murders Zod. The Phantom Zone is literally right there. "But in the movie..." Yes, in the badly written movie they did something dumb so couldn't or wouldn't use the Phantom Zone. So don't make that mistake in the first place. Zod lives, Superman now has a long-term nemesis trapped in another dimension looking for revenge. You know, instead of Superman, Kal-El, the Last Son of Krypton, being a murdering anti-hero.
Took longer to type that than think about it. I'd watch that movie a few more times. Huge fan of superheroes and I don't really even like Superman. He's boring. But undermining him is not the way to refresh the character.
