TL;DR
Money for long-term borrowing moves freely across borders, so liquidity is a global phenomenon. QE bought bonds from asset holders and paid in cash. With this cash they bought other assets, everywhere and in every class. Asset prices rose worldwide, including in countries that never ran a QE programme of their own.
That was inflation, in a market the consumer price index does not cover. And bidding an asset's price up is bidding its yield down, so the same event was a global fall in borrowing costs.
Most econometric models that estimate the neutral rate are fitted to consumer prices, output and interest rates. Asset prices are not among the inputs. So those models interpreted the persistently lower borrowing costs as evidence that the neutral rate itself had fallen. Policy rules then took that estimate as an input, and naturally recommended something close to the policy already in place.
None of this began in 2009. Bernanke saw low rates in 2005 and identified a global savings glut as the cause, favouring a structural explanation over a liquidity one.
This one story does the work of several. It explains why asset prices boomed where no programme ran, why consumer prices stayed quiet and then surged in 2021 when the same mechanics reached households instead, and why every country's estimate fell together despite different demographics, deficits and growth.
What the models report is the rate neutral for consumer prices. The rate that would have kept borrowing serviceable sat far above it for a decade. That is not a measurement problem. It is a specification problem, and a specification problem returns a confident number to the wrong question.