But since, once you spent the $10, they are now somebody else’s income, namely for the person/company you bought from. Which means that now, by simply summing everyone’s incomes, you have an accurate count of GDP, without ever considering how individuals spend their money.
I think the main problem of contention is, confusion about personal income v. national income, the latter of which includes institutions and is the one being used for GDP calculations. Money cannot be spent without being received by anyone, and the moment it is, GDP goes up in the country it’s received in.