The FED prints money and passes this money on to govt via buying US debt (trasury bonds). Once the govt has the money it needs for its huge budget, it spends that money on defence,schools,infrastructure,public services,construction,Fed mortgage,to pay salaries of govt employers, etc. So all that printed money eventually makes it back to the people and causes real inflation that should be 10%. This is why dollar has declined somewhat in proportion to the money supply inflation, but the core CPI hasn’t reflected that. So I am not sure about your original point, the money printed by FED does reach the public eventually.
Now, the core CPI hasn’t moved by 10% because it excludes housing, stocks, energy, food, etc. On top of that, China/Japan artificially set the price of their currency to protect their exports to the US thus keeping the price of most everyday items in walmart, from rising too much. Also productivity has increased tremendously thus keeping up the supply of everyday items, to the money supply.
However, this kind of inflation then shows its face by resulting in asset bubbles in asset markets where supply is limited (real estate, gold, oil,stocks,etc) but demand keeps rising with increasing money supply. Inflation in these assets then forms a bubble as speculators/hedgies/flippers join in. This is why you have seen the price of houses/stocks/gold/oil jump up by huge amounts in last 4-5 years but the core CPI hasn’t been able to reflect that. This is also why I believe that there is a solid ground beneath house/stocks/gold/oil prices. How can you encourage people to save money in such fiscally iresponsible policy? If I get a car for $10K at 5% APR but am getting 20% return from my fund/house then why should I not keep borrowing and buying?