I don't even understand what an "unmodifiable blockchain" held by a single private entity could mean.

They

- have the data of yesterday

- have backups of the data of yesterday

- are going to write today's data on top of it to continue the chain

(insert magic here)

- somehow we are super sure, tomorrow, that today's data has not been tampered with

If somebody can shine some light on the magic part, I'm interested

The point isn't that you can't add bad data: obviously you can.

The point is that if the hashes of the blocks are published, you can't later go back and tamper with that data.