Even if a rich person reinvests everything, the control over large amount of money is what makes it problematic.
Also the idea that welfare doesn't go to investments is wrong. When you buy groceries (or anything really), there is a decision made by the management of the company you buy these things from to reinvest part of it to maintain or build productive capacity.
There is no need for a "capitalist" (owner of the enterprise) to insert themself into the process, they are useless middlemen who get a cut, essentially. (They are not so useless when they do actual managerial work, but then they can be just an employee like everyone else.)