It will only be marginally better, or perhaps worse, having the govt run the GSEs directly.
More to the point, why do we have any govt involvement in more than maybe 10% of the housing market? All it does is distort prices, or at the very least price distortions are a massive unintended consequence.
What we need is much smaller GSEs (and FHAs etc). How exactly to manage them is an important but secondary issue.