"The thing is, the S&P 500 is clearly hugely profitable, so why doesn't the government invest in a wide shotgun approach to fund half (or more) of the entire S&P 500 and in return take half (or more) of the profits to either keep as revenue or reinvest in the S&P 500 or in other stock indexes?"
Gov't investment in any industry just because it's profitable is a bad idea for obvious reasons.
People who defend the huge profit amounts of pharma companies (which ultimately come from consumers needing healthcare, paying either directly or through insurance) say it's fair because of the huge amount they invest to create the drugs, if the government does most of the investing then they can take most of the profits to subsidise free pills for their population. Seems to me like a more direct benefit (in addition to potential economic benefits of boosting investment levels in a high-exporting industry too) than just the government buying S&P500 index funds.
Why don't we go after the actual 'hugely profitable' companies if that is the goal.
To bring it back to the specifics of the original argument, and refine it a bit, the relevant point is, "If these are such easy profits from doing nothing, why doesn't the government get in on it?" Alternatively, why don't the universities themselves develop drugs to the point of FDA approval and mass marketing?
The unfortunate answer is: it's not actually easy profits. It's highly speculative and difficult work to turn a drug from "supported by university research" to "ready to market to the public". Sometimes universities do try to cover some of that ground though.