I think Jon should follow his dad's moral code if he's to wear the S. Superman isn't just super in his powers. He's the hero amongst heroes. A symbol for all Mankind. He's meant to be better. He was sent here to guide and be an example.
That S means something more than just hero and Jon has a duty to uphold what it's come to represent.
I'm not that versed in the superlore but the above is how I've interpreted the character. I could be wrong.
Superman's responsibility and role is much much bigger than say batman. Just the sheer scale of Superman's power demands that he's morality [at least as perceived by the world] is higher.