When you say the government’s printing of money does not go into the hands of the public or producers, but it does do exactly that, just not directly.
When the Government prints money, where does it go ?? It gets deposited into banks, especially those large Wall Street Bank who are shareholders of the Federal Reserve.
What do the Banks do with all those deposits ??? Banks make money by loaning it out, now days they also make money by trading it on Wall Street thru stock purchases, bond purchases, etc.
When Banks loan money that is deposited from the Federal Reserve they do so at much lower rates than they would normally do because they have so much of it to loan.
Lower Interest rates encourage borrowing by Consumers, Businesses, etc....
Consumers get lower car loan rates, home mortgage rates, etc. which increase their consumption, more cars and houses get sold, the price of each starts to rise which is the definition of inflation, print enough money and you get what we are experiencing right now, keep printing even more money and you get Weimar Republic style hyperinflation.
Here’s a short article about the explosion of the money supply it the USA over time.
https://levels.io/inflation-chart/
Either way, the money isn’t just given away, it’s borrowed and there’s some sort of note or bond securing it.
It’s obvious that neither increasing the money supply nor deficit spending alone causes inflation. Just look at the last four decades with big increases in both the money supply and deficit spending yet very low inflation.
Now if the government borrows money and gives it to consumers, as they did with the stimulus payments, they can stimulate a lot of demand and if the supply of goods isn’t there you get inflation.
The short answer to your question is you get the money by borrowing it, by your employer borrowing it and paying you, or by transfer payments from the government.