This discussion goes way beyond an "absolute layman explanation to homotopy type theory" ...
Anyway, I disagree that the "whole idea of type theory is that things can have more than one proof". UIP is about identity types and their proofs. Identity types are very special types. It is far from obvious that they should be inhabited by more than one proof. Indeed, it was a surprising breakthrough when Streicher/Hoffmann discovered that UIP does not hold in general and new axioms like Streicher's K or Univalence needed to be formulated to deal with this question.
It is wrong to say that it's key novelty is that "Homotopy type theory: extends type theory with proofs of equality in a proper way". This was done already with Martin Lof's type theory, as used in e.g. Agda. Homotopy type theory adds univalence!
What is the state of the art of the research on the computational interpretation of univalence? What's working, what's not working? I seem to remember that A Jeffrey said not too long ago that he had a working interpretation.
Anyway, I disagree that the "whole idea of type theory is that things can have more than one proof". UIP is about identity types and their proofs. Identity types are very special types. It is far from obvious that they should be inhabited by more than one proof. Indeed, it was a surprising breakthrough when Streicher/Hoffmann discovered that UIP does not hold in general and new axioms like Streicher's K or Univalence needed to be formulated to deal with this question.