Equal productivity would just mean the average and median wages would be the same.
Government spending, funded by taxes, contributes to the GDP like any other spending.
That's exactly the problem. You're thinking of Gross Value Added (GVA), which strictly talks about value added through normal economic activity. Once you start "removing" productivity through taxation and "adding" productivity back through spending/subsidies you have a new variable- the presence of that variable is the difference between GVA and GDP. That variable doesn't necessarily net out to 0 either, or even close, if the government runs a surplus or deficit as the United States does.
Regardless, GDP can be calculated by summing all income across all sources. It represents a nationwide increase of wealth. The gap between wage per worker and GDP per worker indicates that, despite labor being the only source of wealth generation, labor only receives a fraction of its productive efforts back as wages. I like it because illustrates the gap between capital and labor, as an alternative to the more traditional income inequality between workers.
Its market cap divided evenly among its half million full and part time workers would make all of them millionaires.