Superman is much more difficult to write a good story for than batman. Even though I way prefer superman as a character, my comic book collection has many more batman issues than superman because most superman stories are fairly lame, poorly thought out and try to match him up and with an equally impossibly strong villain to punch.
Writers for superman seem struggle to come up with any meaningful threats and situations for him to tackle. How do you make an invulnerable god vulnerable ... the answer has to be through his humanity. Man of steel did get that right partially having him make a morale decision to choose between Krypton and his adoptive planet Earth, that was a good idea.
I agree that superman is a lot like Captain America in that both always strive to do the right thing, they each have a strong moral compass and will not compromise their beliefs (will never cross that line). Cap is easier to write though because he is 'just' a super soldier and has physical weaknesses. Whereas superman is only truly vulnerable because he cares for people, for Jimmy, for Lois and yes 'Martha' 'Why did you say that name!'. You don't threaten superman, you threaten his world, his adopted people, his values and his family. Superman is best when he has an intelligent villain.
As mentioned, I think Man of steel did get a lot of this right but it struggled with the following aspects:
(1) Kal-el was too slow to show his love for humanity (it was hard to buy);
(2) it needed to have him save people before the final showdown with Zod.
(3) he should not be questioning why he should help humanity, it is who he is (perhaps beyond Jonathan and Martha teaching him right from wrong as a kid). Instead have others (like Lois or reporters) question why he does it when the world as a whole and the media seem so ready to condemn him as a threat like the other kryptonians. What makes him do it, why is he different to them? We know why as the audience, but the people in Zac's cynical DC world don't.
(3) It didn't raise the stakes enough in the final battle with the actual people that Clark Kent cared about i.e. Lois, his mother - the threat never felt credible to them directly. They are a bigger weakness to him than kryptonite. Zod (as a genius general) and should have looked to exploit this weakness more.
(4) as others have said, have superman actively trying to save people during the final showdown with Zod at the expense of his advantage in the clash or even to his detriment.
(5) don't have him choose to kill Zod! Even if it is the logical option that everyone else would take in that situation, Superman would find a way to do it differently.
The death and birth of superman is (in my opinion) a poor superman arc, as is the dark knight returns (a good one to show an unhinged oldman bruce sure but not much of a showcase for superman). Both very poor choices as follow ups to the man of steel and i suspect that these were dictated to Snyder rather than him chose to re-tell them so soon.
