The entire premise of capitalism is that money that gets invested into useful purposes
No, the entire premise of capitalism is people own stuff. Period.
There is an assumption people might invest in useful stuff and make rational decisions in their own best interests. The reality is not quite the same.
When billionaires buy multi-million dollar yachts and diamond crusted iPhones you get to see why kings periodically get their heads chopped off.
The producers add their tax burden to the cost of goods. The study from Harvard econ. sets the price of goods as 22% higher (average) than they would otherwise be without the income tax.
And, if you didn't have a government to take taxes and do the things the public needs, your society would be a shitty place to live, and would be the most brutally Darwinistic thing you can imagine. So those 22% lower costs would be offset by a society which is many many times the worse to live in.
This fictional, utopian tax free society would be not nearly as good as its proponents claim it would be.
If you model your economic system on the Ferengi Rules of Acquisition, you will not like the results.