Suppose we have ten thousands banks, and each of those banks has in its possession a thousand bags of metal. Somewhere around eighty percent of the bags are known to contain gold. Somewhere around twenty percent contain lead. These percentages are not hard and fast, but they are roughly good numbers.
Each bag of gold is worth $100,000, and each bag of lead is worthless, so on average there is $80,000,000 dollars worth of metal at each bank. But that is on average, over all 10,000 banks. The bags cannot be opened except over a lengthy and arduous process which takes years.
The distribution of lead and gold amongst the banks is entirely unknown, but it is definitely not uniform. It is not even entirely clear how many bags a given bank has. If I bought up all the bags from a given bank, or even a given cluster of banks, I could easily end up with all lead. On the other hand, I could just as easily end up with all gold. Most likely I would end up with some mix, but the percentages could easily be radically skewed one way or another. Buying up all the bags from a given bank is a crap shoot: an eighty million dollar crap shoot, on average.
On the other hand, if I have a large enough amount of capital to buy up all of the bags from all of the banks, it is perfectly reasonable for me to expect that 70% to 90% of them will contain gold. If I can buy bags on the open market for $20 right now, that represents a very good investment opportunity; only if I can buy substantially all of them. If I can buy them all, it is even well worth it for me to take out a loan to buy them all, because I only have to pay a couple of percent on the loan whereas my returns on the portfolio will most likely be quite a lot higher. It is a surer bet than (say) buying a house with a mortgage. If I only have enough capital to (say) buy up the bags of one bank, on the other hand, I am basically rolling the dice on my own solvency.
In a nutshell, that is why the federal government can afford to buy up these assets as a responsible transaction, effectively not costing the taxpayers a thing, and far smaller market players are incapable of doing the same thing.
In a peashell, size does matter.
And notice, please, that this does not even take into consideration any salutary external or strategic effects; for example, say, preventing credit lockup and halting a financial downturn on the order of the Great Depression.