Which is, of course, what we currently have, except it's the businesses who are able to "conveniently forget to mention" instead of the borrowers.
I pay myriad bills on time and in full every month, but only three of them--all loans--report my positive payment history to credit bureaus. Yet all of them, from mobile phone providers to landlords to utilities to insurance companies, insisted on being able to look at that payment history. They get the benefit of being able to evaluate me on an incomplete set of data; why shouldn't I get the return?
(The answer as always is that we don't have any leverage. Simply saying "well, then don't use those companies" is a non-answer when every company does it the same way and all of them lobby government to keep it that way.)