How does buying $500k worth of stock ("investing the money") stimulate the economy? Haven't I just purchased a paper asset from the seller? I guess you could argue that the seller who liquidated their position is now free to spend that cash on something productive.

That’s the argument for buybacks and dividends. If companies don’t have any NPV positive projects to take then the cash must go out to shareholders who do.

Aside from the fact that companies need investors there is also the question of investing in IPO's etc.

As far as investments, I'm no economist, but I'm skeptical that the money spent purchasing securities has an equal impact as someone buying goods and services locally. It seems like a disproportionate amount would go right back to other members of the investor class, who are going to use it on other securities, or else high-margin luxury items, and very little of that is actually going to escape that bubble.

I think you could argue that replacing benefits with direct income is still a "win" for society. If they're able, it's more efficient for people to meet their own needs than to be provided for through a bureaucratic process with lots of administrative overhead.