I suspect by "note what it leaves out" you are referring to, e.g. unpaid work like caregiving and household labor, as well as inequality and environmental costs.

I would point out that GDP also has shortcomings in that it does not measure well-being directly (happiness, mental health, life satisfaction) nor does it account for non-economic quality of life factors like political stability, personal freedom, safety or social cohesion.

It's a simple metric to calculate, it's also gamed a lot exactly because of this assumption.

It's extremely faulty to measure general living standards, a country with expensive healthcare will generate higher GDP while having a sicker population, the same repeats for any essential service to quality of life which is fraught with middlemen, each step in the chain increases GDP. Also for shoddy construction, repairs and renovations will increase GDP.

Using GDP as a proxy for living standards is very poor.

Often yes, but I think we're seeing for years now that in US at least, the GDP numbers for a few years don't really match the living standard as felt by the population. That makes it a less than useful metric for purposes of measuring prosperity.