I reckon that's now how SSI solutions use a blockchain (or at least the ones I know of). AFAIK, a blockchain is only used to resolve a DID (an identifier mapping a random ID to a bunch of public keys you control, along with a URL to communicate with your device). Your personal information, as in your name, phone number, email stay in your device and are only exchanged with the party you want to authenticate against.

I'm curious to know if there's something I misunderstood in this space

Zero-knowledge cryptography allows you to post opaque data to the blockchain and selectively reveal properties of it to chosen people and applications.

For example, a zero-knowledge DID can generate a proof to an age-restricted application that you are over 18, without necessarily revealing your birthdate or any other information about yourself.