What part of buying index funds of public shares in a company (aside from direct investment, IPO or private placement which are not hallmarks of index funds) funds the company?

Thieves steal because they know a fence will take the goods of their hands. The fence will take the goods of their hands, because they know they can sell them on.

People buy into IPOs partially because they know a lively secondary market exists, where they can offload the shares later. Index funds are part of that secondary market.

Just to be clear, I don't think investors in IPOs are thieves. I'm just saying that you can legitimately say that the secondary market financies companies just as much as the primary market does. Perhaps a better example might be farmers selling to food factories selling to retailers selling to me. I never hand money directly to the farmers, but you can still say with a straight face that my purchase of bacon funds the pig farm.