While we're getting the government out of the housing finance market, why don't we go ahead and eliminate the mortgage interest deduction, which acts as a hidden government subsidy (cue anguished screams).
Seriously, many of the problems we are facing came from the practice of selling and securitizing mortgages. If you are the initial lender (or "originator" in the parlance) you have no incentive to have very strict underwriting guidelines and not much incentive to rigidly enforce the guidelines you do have, because you will immediately sell the loan to an investment bank who will put together a pool and issue RMBS's (residential mortgage backed securities). The investment bank doesn't have that much incentive to scrutinize the deals either, because the economic risk of default is being passed on to the RMBS investors. By contrast, in the days prior to widespread securitization, "Podunk Building & Loan" would keep that loan on its books until final payment. Under those conditions, Podunk had every incentive to set and maintain strict underwriting standards. The government could help to set the current system aright by passing laws to eliminate mortgage securitization, or at least to reign in some of the worst practices. Samclem is right that when the economic loss falls on the person who was sloppy, that person will clean up his act. However, the current separation of the origination process from the economic loss prevents this type of market driven self correction.