I am in the process of completing a home mortgage. The process has become awful again, not as bad as it was in the early 90's but harder and more frustrating than in the mid-2000's. There is one set of rules, and if one's situation does not fit those rules, good freaking luck. Right now, for example, getting a home mortgage when one is self-employed, even in a pretty large business with a decade of history, is really hard.
At the moment there is nobody doing conforming mortgages except Fannie and Freddie. Indeed there is almost nobody doing mortgages of any kind except Fannie and Freddie. If the free market wants the business they can have it. (They just don't want it at this sort of interest rate spread - and I don't blame them.)
All the government need to do is tell Frannie to raise their price a little each quarter. Currently they charge 20-25bps for guaranteeing mortgages. (The free market won't take credit risk at that price.) So it is entirely open to the FHFA (and hence the Treasury) to tell Fannie and Freddie to raise their prices by 5bps. The government will get paid better for the risk they are taking (and what free market ideologue will disagree with that) and the private sector can compete if they want to.
I doubt the free market will. But then in a quarter or two Frannie can raise their pricing by another 5 bps. And a quarter or two later Frannie can raise by another 5bps.
At some stage you will get to a level where the private sector chooses to compete. Frannie should not set its price competitively though. In another quarter they should raise the price another 5bps. And in another quarter they should raise again.
By the way, this is a classic example of not learning from your last mistake. That spread is absurdly low. I wouldn't guarantee my best friend's loan for 20bp. Would you take on the default risk of a $100,000 mortgage for $200 a year?