There's no disagreement here. FDIC/SIPC protect against insolvency. The context of this discussion is borrower credit risk, i.e., the risk that the borrower becomes insolvent.
Bank accounts are FDIC insured up to $250,000 each.
If your broker lends out your stock to short sellers, it will always return your shares, even if the short seller gets margin called and doesn’t have the money to pay back their broker.
I’m not sure what you mean by “index fund”, but securities/stocks are protected by SIPC insurance, up to $500,000 per account. You will get your stocks back if a brokerage fails.