Yes, but to assume that government will accomplish this better than private markets is to truly live in Oz. Private investments will increase the GDP which will need to be divided up to provide benefits. Which increased GDP more, the money invested to create Microsoft or the identical amount spent on welfare, the military or a study of cow flatulence?
If we can all invest our own money don't you think that will increase GDP more than the government holding the money and then paying it out at a later date, unless they decide to cut the benefits, which they can do.
My best guess is that the opposite is probably true. People bidding up the stock market will have verey little effect on the GDP; if SS surpluses are used to retire the national debt, we would benefit immeasurably, in several forms, one of which would be higher future GDP.