It is very often contracted, even erroneously.

• What's it all about, Alfie?
• What's It? - The Award Winning Game Where Creative Minds Think Alike!
• Sun, sea and silver service: what’s it like crewing on a superyacht?
• What's It? - Information Today
• What's It Like on the Pope's Plane?
• What's it to you, anyway?
• *What's it do?"

The problem with your example is that in the short, stand alone phrase "What is it?" (just as with Who is it?), the emphasis is on is, not what. If there is no emphasis on is, then the phrase is simply What? If the emphasis is on it, then something for the dummy-it must be stated, as in the cases above, or, What's it like outside? (Who's it gonna be?)

I hope this qualifies for "a cogent, simply described explanation of why the verb BE in" what is it isn't contracted.

It's a perfectly grammatical contraction, though it doesn't commonly stand alone; it's usually followed by another word.

• What's it all about, Alfie?
• What's It? - The Award Winning Game Where Creative Minds Think Alike!
• Sun, sea and silver service: what’s it like crewing on a superyacht?
• What's It? - Information Today
• What's It Like on the Pope's Plane?
• What's it to you, anyway?
• *What's it do?"

The problem with your example is that in the short, stand alone phrase "What is it?" (just as with Who is it?), the emphasis is on is, not what. If there is no emphasis on is, then the phrase is simply What? If the emphasis is on it, then something for the dummy-it must be stated, as in the cases above, or, What's it like outside? (Who's it gonna be?)

edited to add: Please see @John Lawler's comment.