You cant have D&D realism and such without limiting the gameplay more and taking out the crazy super hero stuff. They are like the antithesis of one another.
What is D&D realism, though? Realism in a fictional setting is build by using a consistent set of rules (and I don't mean combat mechanics, but laws of your fictional reality), which you then apply to whatever story you create. So if you want to explain why a character cannot jump this high, then you need also to explain how, by the same rules, that giant red dragon is flying by flapping its wings, considering the sheer body size and shape. Why doesn't it look like a pterosaur instead?