GDP is not productivity. If I manage to produce/sell a hundred gizmos for $1 each, displacing my competition who were producing/selling 50 gizmos for a price of $5 each, I just halvened gdp produced even though both me and people using the gizmo are more productive than before.
I think precisely the point is that money isn't wealth.
Government can print as much of the latter as they want; wealth goes up when we can collectively buy more and down when we can collectively buy less, regardless of how many dollarpounds that is.
But GDP is measured in money, and can only connect to wealth if we get get inflation right, but that's really hard because inflation depends on what you want to buy — childcare costs don't matter if you have no kids of that age.
That said, I trust the domain experts to get this right, even though the various governments may be incentivised to claim their own preferred numbers. Even at worst, they'll have thought of vastly more influences than I can even imagine.
