>> New money that is anticipated will lead to anticipated price rises. So it doesn't matter (too much) who gets first dibs.
There is a significant delay before the newly created money affects wages.
Shareholders benefit from the new money instantly (since the market reacts to it quickly and it drives up the value of their stocks) but wage earners won't benefit from the new money until they get their annual 2% salary raise at the end of the year.
That delay is significant because it allows shareholders to compound their ROIs several times before wages catch up.
The Fed mostly buys government bonds, the markets for which are very liquid and efficient.
(The government however does benefit from the Feds buying their bonds, instead of injecting money into the system some other way.)