I think one of the biggest downfalls is that every dolar is weighted the same, so money that's wasted is counted just as heavily as money spent to improve "productivity and economic health". $1MM worth of missles that are burried in the dessert effect GDP equally as $1MM worth of food or vaccinations. Also, (although this wasn't specifically addressed in your questions) GDP doesn't speak to the distribution of wealth or economic stability, so coutries like Saudi Arabia can have a really high GDP, but not a really great standard of living.
GDP: A Meaningless Statistic « Irish Liberty Forum
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