Wouldn't the value be calculated as whatever the government payed for them just like the rest of GDP? Maybe I'm misunderstanding how government spending is adde

Wouldn't the value be calculated as whatever the government payed for them just like the rest of GDP? Maybe I'm misunderstanding how government spending is added to GDP but the fact that the amount spent on something doesn't always correlate to the value doesn't seem to be a problem unique to the government and hardly seems a reason to ignore what is in the US up to a few trillion in spending