Sheila Blair basically said that our financial system rests on debt because debt is collateral for banks. In 2008, when the market went belly up, the Fed decided that it's goal was going to be that it would put itself in a position to be able to pay it's greatest debts, and thus smashed down the interest rates to near zero in order to buy corporate junk bonds to keep the market afloat. Lowering the interest rates also supported the mortgage market--not individuals but the market. Even if individual home owners couldn't afford to stay in their homes, someone else could capitalize on the situation and buy the house.
So what would have happened (or what would happen) if the Fed buys all the student loans (maybe instead of or in addition to buying corporate junk bonds)? It would go against future payments, but it would cost tax payers nothing at this time. We just wouldn't get any future payments on student loans (but those people would still be tax payers). As
@Hemotep pointed out, many studies suggest that student loan forgiveness would have almost no downside when it came to individuals and only had minimum effect on inflation. The reason we have inflation right now is because of very real supply issues, so no amount of financial leveraging by the Fed is going to change the fact that we have a shortage on baby formula. There is no actuarial model that can create microchips.
The problem with canceling out student debt is that it would show the American populace at large that we can cancel massive amounts of debt and make people's lives better, more people will start asking for debt to be canceled. First, it will be medical debt, and then, people will start arguing that consumer debt, especially for things like food, might also need to be scrutinized. Canceling huge chunks of debt for a generation of people, would show that generation of people that you don't always actually have to pay your debts, because we have decided as a society that the level of debt the average American has in order to survive is too high, and we need to stop the rent extraction. 98% of all wealth that is created is really just debt. The reason these stores want you to open a line of credit, is because they can say that is wealth creation--because they have created a line of credit for you to pay them monthly. If you cancel all of this debt, then you are ultimately canceling this wealth. So the only people that end up losing here, are those people that hold all the debt. In the past, we have had this problem in society--where people are too indebted to live happy lives. The ancients were able to quell the rabble by having debt jubilees where they did just go and forgive everyone's debt. In recent history, all of our wealth redistribution has been violent. I guess we know which way our leaders want this to go.