those institutions are not lending your money out in uncollateralized loans.

Yes they are, banks operate on fractional reserve. Banks do signature loans all the time. A popular one is known as a credit card.

https://www.investopedia.com/terms/f/fractionalreservebankin...

It does not protect if the asset invested in goes down in value.

FDIC is protection against the bank being insolvent.

SIPC does not protect customers against losses from the rise and fall in the market value of investments.

https://www.sipc.org/for-investors/what-sipc-protects