That makes the task of paying off a home mortgage, or a government debt, very difficult.

Why would we want to pay off the public debt? Indeed, the fact that you've compared the public debt to a mortgage implies that you don't appreciate the fact that the government issues the currency that denominates the debt, while the same cannot be said for a household.

I believe the Federal reserve receives these interest payments. This represents a huge income for private banks to give the government the liberty to borrow money as it pleases.

The Federal Reserve is part of the government; they even say so themselves. Indeed, the Fed's profits (primarily from holding so many bonds) are actually paid back to the Treasury.

I really can't see the point of having the private banks in the middle to receive this interest. What does this accomplish other than the enrichment of private corporations?

That depends on what you mean by that. If you mean, why do the banks serve as intermediaries between primary auctions and open market operations/repurchase agreements, then the answer is, because of misguided notions that primary auctions represent a "market test" for government debt, and that having the Fed buy bonds directly from the Treasury would be "the central bank financing public deficits," which we've been assured is bad for some reason.

If you mean, why do we pay interest on public debt to private banks, then the answer is that we choose to use liquidity management to maintain the interbank lending rate, which requires that there be a risk-free cash alternative that has the same yield as the desired opportunity cost of cash.