Fannie and Freddie were a major cause of the crisis but not through the mechanism you describe. They never purchased individual subprime loans and sold them as bundles (or at least not at any scale that matters.) They bought credit-worthy loans, packaged them and sold them - and those loans by and large performed okay.

Where they got into trouble was when they tried to fulfill their mandate to support the sub-prime market. Instead of buys, bundling, and then selling the bundles, they decided instead to buy the "AAA" tranches of bundles of loans that other people had already securitized. They kept these loans as their own assets (unlike when they sold the other loans off.) These "AAA" subprime tranches were not really "AAA" quality as we all found out - but the key point here is that Fannie and Freddie were NOT selling subprime mortgages to the markets. They were BUYING the AAA tranches of subprime pools - to the tune of trillions of dollars. They were doing this for two reasons - first, they thought they could make a lot of money for their shareholders. That is, it was an investment strategy - just one that went horrendously wrong. Second, they were responding to the requirement that they purchase up to about 40% of all loans made to underserved groups that was required in the 2005 National Affordable Housing Act Goals.

So if they were BUYERS of the bad loans as opposed to being the SELLERs, what happened? Well, because the loans were on their books when the loan started defaulting and the market for subprime went south in early 2008, they had to mark down those assets on their books. This mean that they had to use up their equity to balance their books. As the market prices continued to deteriorate the capital markets started to question whether Fannie and Freddie could honor their obligation to guarantee the "prime" mortgage backed securities that the usually issued. As the markets became convinced that they did not have the assets to do that, the government - which has always "implicitly" guaranteed fannie and freddie, decided to step in and take them over. That is when taxpayers became liable for the losses at fannie and freddie. It was because of the investments they made in subprime, not because they were issue subprime.