What makes you think rich people don’t invest? Besides a lot of the wealth is from shares of stocks.
Private investeors tend to be completely short-term and profit-oriented. While that is fine in certain situations, there also needs to be a longer-term, strategic actor.
For example, I very much doubt that private interests would ever have funded China's high-speed rail network. The huge improvement in connectivity will benefit the country greatly in the long run, but it would not have happened if billionaires were solely responsible for the country's economic development. We know this because the US's overwhelmingly private economy has built exactly zero miles of HSR.
Some billionaires are fine, but in my opinion they must never be allowed to dominate the government. The billionaires' short-term and selfish outlook would neglect the country's infrastructure -- as has happened in the US.