That person speaks about the money not taxed being used for taxable economic activity on the same year, as opposed to that same money being spent on government programs, that generate less taxable revenue (e.g. because while paying contractors will generate taxable revenue, hiring a public servant to do the same job won't).
Your reasoning is that the money spent by private people can also be investment, which will increase future gdp growth. However, you don't explain why government spending couldn't increase future gdp growth. Obvious counter-examples being money spent in schools, public research programs, etc.
The comment claims that it isn't possible for lower taxes to increase tax revenues. It's possible whenever the taxpayers would have used the money more productively than the government. Which may not always be the case, but it's certainly possible for it to be the case, and indeed is not an implausible default assumption because governments generally aren't subject to competitive pressure and consequently tend to allocate resources in ways susceptible to inefficiency, waste and corruption.
Your reasoning is that the money spent by private people can also be investment, which will increase future gdp growth. However, you don't explain why government spending couldn't increase future gdp growth. Obvious counter-examples being money spent in schools, public research programs, etc.