Cryptocurrencies create a minimum coverage of ability to conduct financial operations, they are assets that a third party can not freeze, they are assets that a third party can not invalidate. Of course the first use cases will be illegal ones.
Bitcoins can be divided, the smallest fraction is called a "Satoshi" and is a millionith of a Bitcoin (So a cent is 500 Satoshis). The blockchain is a peer-to-peer, which while technically 3rd party, is not something that "you have coin on". The blockchain is basically a cryptographically signed ledger that says a certain public key has this much bitcoins. If you have the matching private key, you have the coins. Exchanges are mostly used to exchange between Bitcoin and some other currency, like US Dollars or another crypto currency.
I mean, I agree with you entirely: that is BitCoin’s unique value proposition. It’s just good to see it out on the table.