Normalising GDP for the amount of money in the system suggests that velocity of money is the is actually the interesting metric.
Say the monetary base is $100 and the GDP is $1,000 initially, then the monetary base doubles to $200 and the GDP doubles to $2,000. It looks suspiciously like no real economic activity was generated and that suggests that velocity is the growth measure.
You're essentially assuming an increase in the money supply leads to inflation, which it does if nothing else is different either, but if you also have e.g. a larger population then you're looking at something else.
Say the monetary base is $100 and the GDP is $1,000 initially, then the monetary base doubles to $200 and the GDP doubles to $2,000. It looks suspiciously like no real economic activity was generated and that suggests that velocity is the growth measure.