And if the people kept the money, their expenditures on goods and services would also be a component of GDP.
Right. If their propensity to consume is 100%. Which it is not. And never has been. Which is why gov consumption is a much more stimulative factor than tax cuts. Tax cuts often go to savings, particularly among the more wealthy. Gov expenditures are stimulative. Private spending is stimulative. Some, in both cases are more stimulative than others.
Over time, cutting taxes has been proven to be less stimulative than government expenditures in most cases. To my knowledge, we have no gov projects that call for digging and then filling in holes. It would be stimulative, but the multiplier would be low. However, if you did what the gov wants to do and developed programs to make or repair infrastructure, it would be quite stimulative.
Basic economics, though not libertarian or austrian economics.
By the way, building or repairing infrastructure is not done by the gov. It is done, of course, by the private sector. The private sector is paid, sector eployees are paid, they buy more stuff, which provides more money to more private sector folks. Which should scratch your itch about private sector involvement. And, it makes economies take off, based on the economic history of this country. Cutting taxes to put more money in the hands of the private sector NEVER does in bad economic times (times of high unemployment).