I would much rather just whack the big players with antitrust law than give their power to some government agency. A worthless mission statement of some bureaucracy is just that, worthless. If said bureaucracy is using the same information Equifax is using to provide the same services the abuse potential is right there and it's probably only a matter of time until it starts getting used to screw whoever the outgroup of the minute is.
What is sorely needed is privacy legislation that says that data about an individual is fundamentally the property of that individual, and that the license for commercially possessing/processing an individual's data can be revoked at any time. Only then can there be competition in this market.
What will having a market of providers for this service do for me that having a crown corporation or federal ministry won't?
Let's look at it the other way. Could you imagine privatizing the IRS. Why would I want that?
I think you understand that the point of taking this course of action is to prevent/reduce these kinds of abuses from happening.
The goal isn't to enshrine abuse of authority in governmental agencies, but to prevent nongovernmental agencies from abusing that authority.