The Fed has been talking out both sides of their mouths. Fed and ECB printed vast amounts of money and much of it is causing this global inflation we're seeing here. They claim it's only about 2-3% pts of the 9% inflation we're seeing but that number is likely to be revised up. You can print that amount of money when supply chains were shuttered and not expect inflation. The US and EU are much of the world's demand generators so where they go, so does the supply of goods and services.
"They", if you mean the feds, actually claim the impact of stimulus on inflation has fallen towards zero. It's completely independent analysts who converge on 2-3 percentage points as the number. I'm not sure why you think they're lying or that they're going to revise that # up when that estimate has been pretty static for over six months now.
And don't know why you dumped the EU in with the USA when the EU didn't do much stimulus at all compared to the USA did yet still saw similar levels of inflation.
Yes, demand went up in the USA and the USA is a big market, but stimulus wasn't the only reason for an increase in demand. With the pandemic a lot of people didn't spend money on travel and ended up buying shyt and getting it shipped to them instead. And people stopped spending money on services and spent it on goods instead because they didn't want to interact with people in public. Thus demand for goods was going to increase at the same time factories got shuttered and supply chain issues got hit, regardless of whether stimulus was given or not.
Besides, demand for non-essential good has been a minor part of inflation. Inflation in energy prices is around 30-40%. Inflation in food prices is around 10%. Inflation in other goods has only been 6% in the USA, and closer to 4.5% in Europe.