Perhaps a special fully transparent non-profit. You can pay employees whatever wage the market demands and with certain limits, like executive packages that have to be approved by "members" (policy holders).
As to your questions:
1 - The gov shouldn't do it because we want multiple of these entities, not just one. We need choice and competition, no single point of failure.
2 - It is common to say that extending home mortgages to higher risk groups, which Freddie played a role, was to further social goals. I simply do not buy it. Everything points to money. Extending these mortgages furthered the goals of the lenders and investment bankers and home builders as well. Look at who gave money to the politicians to make these policies happen. Hint: it was all of the above, except poor people. Yes, people who couldn't afford their homes benefited but they were not the lead players in making these policies happen.
As to your third question, companies that attempt to be more of a coop have to compete in the same market as the for-profit entities. I don't think you can solve this problem without leveling the playing field.
As to your questions: 1 - The gov shouldn't do it because we want multiple of these entities, not just one. We need choice and competition, no single point of failure.
2 - It is common to say that extending home mortgages to higher risk groups, which Freddie played a role, was to further social goals. I simply do not buy it. Everything points to money. Extending these mortgages furthered the goals of the lenders and investment bankers and home builders as well. Look at who gave money to the politicians to make these policies happen. Hint: it was all of the above, except poor people. Yes, people who couldn't afford their homes benefited but they were not the lead players in making these policies happen.
As to your third question, companies that attempt to be more of a coop have to compete in the same market as the for-profit entities. I don't think you can solve this problem without leveling the playing field.