Why isn't "it" used in place of "he or she", "he/she", "s/he" etc.?

I believe that "it" in the case of "it's me" or "it's John" is an expletive. Like Coline Fine commented above, it is a syntactic placeholder, because in English we can't just say "is me/I am" or "is John/John is" (which is fine in other languages, e.g. Spanish). So, dictionary.com is being extremely misleading in the "it's John" example, or even outright wrong, depending on your theoretical point of view.

It's the same expletive-it that shows up when we say things like "it rains a lot here". What rains? The word "it" is here simply because we can't have a verb without some sort of a subject in English; it doesn't refer to any explicit thing.

So, I would say that we don't use "it" to refer to (non-infant) humans, and "it's John" is not a counterexample.

A hundred years ago, in English, the standard gender-neutral pronoun was clearly "he", but in recent years we started finding it inherently sexist. As a result, we are in a wishy-washy period in English where there is a need for a gender-neutral pronoun, but people aren't quite agreed on what it is. In my observation, "they/them/their" is emerging as the clear frontrunner -- certainly this is true in spoken and informal speech.

Example:

"I just got a call from someone at the doctor's office." "Well, what did they say?"

As far as I know, the use of it to refer to a person is only limited to children of unspecified sex, as reported by the NOAD too.

it |ɪt|
pronoun [third person singular]
1. used to refer to a thing previously mentioned or easily identified: a room with two beds in it | this approach is refreshing because it breaks down barriers.
  • referring to an animal or child of unspecified sex: she was holding the baby, cradling it and smiling into its face.
  • referring to a fact or situation previously mentioned, known, or happening: stop it, you're hurting me.
2. used to identify a person: it's me | it's a boy!

Shifting away from he and she in general is out of the question. The idea of shifting away from two very established pronouns seems incredibly daunting. Pronouns form a closed class, and closed classes of words are very hard to modify; these two are used a lot. I would bet a dollar that she and he show up in top 100 most used English words in nearly every study or count. Finally consider that you aren't just changing he and she here, but also the objective and personal pronouns, which makes this even more daunting. That said, the pronouns of English have changed relatively recently. The Old Norse paradigm for they was borrowed; her is one remnant of the old system.

Replacing the the gender-neutral he with it seems to make sense based on the semantics posted in the question, but is probably also delayed by the reasons above. Further, in my use referring to any person as it, even a baby, is considered rude. So there's that hump, too.