The entire idea of government investing in the stock market bothers me, as almost everyone in government has some connection or interest in some sector of the company or some particular businesses. The controls required to protect such a system from corruption would be costly.

Frankly I think social security is too complicated in its attempt to resemble a kind of last ditch retirement plan. It should be replaced by a straightforward old age (obsoletion)/disability insurance program that comes out of regular revenues (might mean raising regular taxes... but it also means no more social security tax... net reduction of taxpayer burden and there would be less quarreling over fairness).
The pay-out for the new program should be one reasonable minimal payment that a person can live simply off of. No relation to previous income. If a person has a big house not yet paid for and becomes disabled or retires without savings... you'll live, but you'll need a smaller house.

The government should provide only a safety net. To maintain your old lifestyle in any form, that should take your own saving and investing.
Of course poor people and those disabled young shouldn't be any worse off than before under such a system... and that's good. They're the ones most understandable not to have savings.
I think most of the resentment is toward middle class folk who spend all their money and then expect to live like they did into retirement.

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"Man lives in the sunlit world of that which he believes to be reality. But unseen by most is an underworld, a place that is just as real... but not as brightly lit... A DARK SIDE!"
-opening from Tales From the Darkside