Suppose I'm rich. The Fed creates a dollar. The Fed offers me a dollar for my treasury bill worth $1. I accept. As a result, I am now a dollar richer but a $1 treasury bill poorer. Have I gained? I have $1 more in money, but $1 less in assets. So I'd say no, I haven't gained. The only way I gain is if the Fed pays me more than the treasury bill is worth. And since the Fed pays market price, this is difficult to do.
Every right-wing commenter in the world would immediately see that this is wealth redistributing socialism and needs to be stopped instantly.
It's the same if you decide to pay rich people for their bonds. Only, you know, it helps rich people instead. You bought a million dollars of bonds last year, this year you can sell them for $1.1 million. The profit goes straight into the pockets of... people who can afford to buy a million dollars of bonds.
I.e. if the bid/ask is $100/$100.5, the fed will have to buy at $100.5, even if the next market trade might have happened at $100.25. So the rich owners of treasuries do reap a small reward, something on the order of the size of the spread.
That said, your point is generally correct. Buying treasuries results in $100 of money injected into the economy, while redistributing only $0.25 of wealth.
Meanwhile raising the market price of bonds makes debt servicing less expensive, saving taxpayers' money. Unless the market expects inflation, lowering bond prices again.
That's not to say there aren't massive gains to society from a policy mechanism that both provides a relatively efficient and reversible means to manage inflation and helps stabilise the banking system, but the fact is it favours the wealthy much more than the alternatives.
The newly created money is worth more when it's new. When the average person finally receives the money, the value is worth less through inflation.
Secondly, the rich can borrow money from the fed at 0%. Then they can buy bonds from the government and receive interest on that. How does the average person do that?
I guess you are vastly more likely to own a treasury bill if you are rich.
If you are poor Fed buying treasury bills does nothing for you.
The Fed is a large enough new buyer that it will move the market price.